id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-21600 | Plane Section of Spheroid Perpendicular to Axis of Revolution is Circle | Let $\SS$ be a spheroid.
Let $\PP$ be a plane section of $\SS$ such that $\PP$ is perpendicular to the axis of revolution of $\SS$.
Then $\PP$ is a circle. | A spheroid is defined as the solid of revolution formed by rotation of an ellipse about one of its axes.
Hence the result by definition of solid of revolution.
{{qed}} | Let $\SS$ be a [[Definition:Spheroid|spheroid]].
Let $\PP$ be a [[Definition:Plane Section|plane section]] of $\SS$ such that $\PP$ is [[Definition:Line Perpendicular to Plane|perpendicular]] to the [[Definition:Axis of Revolution|axis of revolution]] of $\SS$.
Then $\PP$ is a [[Definition:Circle|circle]]. | A [[Definition:Spheroid|spheroid]] is defined as the [[Definition:Solid of Revolution|solid of revolution]] formed by [[Definition:Rotation|rotation]] of an [[Definition:Ellipse|ellipse]] about one of its [[Definition:Axis of Ellipse|axes]].
Hence the result by definition of [[Definition:Solid of Revolution|solid of r... | Plane Section of Spheroid Perpendicular to Axis of Revolution is Circle | https://proofwiki.org/wiki/Plane_Section_of_Spheroid_Perpendicular_to_Axis_of_Revolution_is_Circle | https://proofwiki.org/wiki/Plane_Section_of_Spheroid_Perpendicular_to_Axis_of_Revolution_is_Circle | [
"Spheroids"
] | [
"Definition:Spheroid",
"Definition:Plane Section",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Axis of Revolution",
"Definition:Circle"
] | [
"Definition:Spheroid",
"Definition:Solid of Revolution",
"Definition:Rotation",
"Definition:Ellipse",
"Definition:Ellipse/Axis",
"Definition:Solid of Revolution"
] |
proofwiki-21601 | Sum of General Harmonic Numbers in terms of Riemann Zeta Function | :$\ds \harm r x - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \harm r {x - \dfrac k n} = \paren {1 - n^{1 - r} } \map \zeta r$ | === Lemma ===
{{:Sum of General Harmonic Numbers in terms of Riemann Zeta Function/Lemma}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \harm r x - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \harm r {x - \dfrac k n}
| r = \sum_{j \mathop = 1}^\infty \paren {\frac 1 {j^r} - \frac 1 {\paren {j + x}^r} } - \dfrac 1 {n^r}... | :$\ds \harm r x - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \harm r {x - \dfrac k n} = \paren {1 - n^{1 - r} } \map \zeta r$ | === [[Sum of General Harmonic Numbers in terms of Riemann Zeta Function/Lemma|Lemma]] ===
{{:Sum of General Harmonic Numbers in terms of Riemann Zeta Function/Lemma}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \harm r x - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \harm r {x - \dfrac k n}
| r = \sum_{j \mathop = 1... | Sum of General Harmonic Numbers in terms of Riemann Zeta Function | https://proofwiki.org/wiki/Sum_of_General_Harmonic_Numbers_in_terms_of_Riemann_Zeta_Function | https://proofwiki.org/wiki/Sum_of_General_Harmonic_Numbers_in_terms_of_Riemann_Zeta_Function | [
"Sum of General Harmonic Numbers in terms of Riemann Zeta Function",
"General Harmonic Numbers",
"Riemann Zeta Function"
] | [] | [
"Sum of General Harmonic Numbers in terms of Riemann Zeta Function/Lemma",
"Sum of Absolutely Convergent Series"
] |
proofwiki-21602 | Conservative Force gives rise to Two Forms of Energy | Let $E$ be a energy which has come about as a result of a conservative force.
Then $E$ comes in two forms:
:$(1): \quad$ Kinetic energy
:$(2): \quad$ Potential energy. | {{ProofWanted|don't even know what this means}} | Let $E$ be a [[Definition:Energy|energy]] which has come about as a result of a [[Definition:Conservative Force|conservative force]].
Then $E$ comes in two forms:
:$(1): \quad$ [[Definition:Kinetic Energy|Kinetic energy]]
:$(2): \quad$ [[Definition:Potential Energy|Potential energy]]. | {{ProofWanted|don't even know what this means}} | Conservative Force gives rise to Two Forms of Energy | https://proofwiki.org/wiki/Conservative_Force_gives_rise_to_Two_Forms_of_Energy | https://proofwiki.org/wiki/Conservative_Force_gives_rise_to_Two_Forms_of_Energy | [
"Energy",
"Conservative Forces"
] | [
"Definition:Energy",
"Definition:Conservative Force",
"Definition:Kinetic Energy",
"Definition:Potential Energy"
] | [] |
proofwiki-21603 | Category of Locales is Category | Let $\mathbf{Loc}$ denote the category of locales.
Then:
:$\mathbf{Loc}$ is a category | By definition, the category of locales is:
:the dual category of the category of frames
From Category of Frames is Category:
:the category of frames is a category
From Dual Category is Category:
:the category of locales is a category
{{qed}} | Let $\mathbf{Loc}$ denote the [[Definition:Category of Locales|category of locales]].
Then:
:$\mathbf{Loc}$ is a [[Definition:Category|category]] | By definition, the [[Definition:Category of Locales|category of locales]] is:
:the [[Definition:Dual Category|dual category]] of the [[Definition:Category of Frames|category of frames]]
From [[Category of Frames is Category]]:
:the [[Definition:Category of Frames|category of frames]] is a [[Definition:Category|catego... | Category of Locales is Category | https://proofwiki.org/wiki/Category_of_Locales_is_Category | https://proofwiki.org/wiki/Category_of_Locales_is_Category | [
"Category of Locales"
] | [
"Definition:Category of Locales",
"Definition:Category"
] | [
"Definition:Category of Locales",
"Definition:Dual Category",
"Definition:Category of Frames",
"Category of Frames is Category",
"Definition:Category of Frames",
"Definition:Category",
"Dual Category is Category",
"Definition:Category of Locales",
"Definition:Category"
] |
proofwiki-21604 | Category of Frames is Category | Let $\mathbf{Frm}$ denote the category of frames.
Then:
:$\mathbf{Frm}$ is a metacategory | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory.
For any two frame homomorphisms their composition (in the usual set theoretic sense) is again a frame homomorphism by Composite Frame Homomorphism is Frame Homomorphism.
For any frame $L = \struct{S, \preceq}$, we have the identity mapping $... | Let $\mathbf{Frm}$ denote the [[Definition:Category of Frames|category of frames]].
Then:
:$\mathbf{Frm}$ is a [[Definition:Metacategory|metacategory]] | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a [[Definition:Metacategory|metacategory]].
For any two [[Definition:Frame Homomorphism|frame homomorphisms]] their [[Definition:Composition of Mappings|composition]] (in the usual [[Definition:Set Theory|set theoretic]] sense) is again a [[Definition:Fra... | Category of Frames is Category | https://proofwiki.org/wiki/Category_of_Frames_is_Category | https://proofwiki.org/wiki/Category_of_Frames_is_Category | [
"Category of Frames (Lattice Theory)"
] | [
"Definition:Category of Frames",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Frame Homomorphism",
"Definition:Composition of Mappings",
"Definition:Set Theory",
"Definition:Frame Homomorphism",
"Composite Frame Homomorphism is Frame Homomorphism",
"Definition:Frame (Lattice Theory)",
"Definition:Identity Mapping",
"Identity Mapping is F... |
proofwiki-21605 | Equation of Epitrochoid | Let a circle $C_1$ of radius $b$ roll around the outside of another circle $C_2$ of radius $a$.
Consider a point $P$ on the line of a radius of $C_1$ at a distance $d$ from the center of $C_1$.
Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.
Let $P$ be a point on the circumference ... | :600px
Let $C_1$ have rolled so that the line $OC$ through the radii of $C_1$ and $C_2$ is at angle $\theta$ to the $x$-axis.
Let $C_1$ have turned through an angle $\phi$ to reach that point.
By definition of sine and cosine, $P = \tuple {x, y}$ is defined by:
{{begin-eqn}}
{{eqn | l = x
| r = \paren {a + b} \co... | Let a [[Definition:Circle|circle]] $C_1$ of [[Definition:Radius of Circle|radius]] $b$ roll around the outside of another [[Definition:Circle|circle]] $C_2$ of [[Definition:Radius of Circle|radius]] $a$.
Consider a [[Definition:Point|point]] $P$ on the line of a [[Definition:Radius of Circle|radius]] of $C_1$ at a dis... | :[[File:Curtate-Epitrochoid.png|600px]]
Let $C_1$ have rolled so that the [[Definition:Straight Line|line]] $OC$ through the [[Definition:Radius of Circle|radii]] of $C_1$ and $C_2$ is at [[Definition:Angle|angle]] $\theta$ to the [[Definition:X-Axis|$x$-axis]].
Let $C_1$ have turned through an angle $\phi$ to reach ... | Equation of Epitrochoid | https://proofwiki.org/wiki/Equation_of_Epitrochoid | https://proofwiki.org/wiki/Equation_of_Epitrochoid | [
"Epitrochoids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Point",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Point",
"De... | [
"File:Curtate-Epitrochoid.png",
"Definition:Line/Straight Line",
"Definition:Circle/Radius",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Sine",
"Definition:Cosine",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Arc Length of Sector"
] |
proofwiki-21606 | Recurrence Relation for General Harmonic Numbers | :$\harm r x = \harm r {x - 1} + \dfrac 1 {x^r}$ | {{begin-eqn}}
{{eqn | l = \harm r x
| r = \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + x}^r} }
| c = {{Defof|General Harmonic Numbers}}
}}
{{eqn | r = \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + x}^r} } - \dfrac 1 {x^r} + \dfrac 1 {x^r}
| c = a... | :$\harm r x = \harm r {x - 1} + \dfrac 1 {x^r}$ | {{begin-eqn}}
{{eqn | l = \harm r x
| r = \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + x}^r} }
| c = {{Defof|General Harmonic Numbers}}
}}
{{eqn | r = \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + x}^r} } - \dfrac 1 {x^r} + \dfrac 1 {x^r}
| c = a... | Recurrence Relation for General Harmonic Numbers | https://proofwiki.org/wiki/Recurrence_Relation_for_General_Harmonic_Numbers | https://proofwiki.org/wiki/Recurrence_Relation_for_General_Harmonic_Numbers | [
"General Harmonic Numbers",
"Recurrence Relations"
] | [] | [] |
proofwiki-21607 | General Harmonic Number Additive Formula | :$\ds \harm 1 {n x} = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {\harm 1 {x - \dfrac k n } } + \ln n$ | {{begin-eqn}}
{{eqn | l = \map \psi {n z}
| r = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \map \psi {z + \frac k n} + \ln n
| c = Digamma Additive Formula
}}
{{eqn | ll= \leadsto
| l = \paren {-\gamma + \harm 1 {n z - 1} }
| r = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {-\gamma + \harm 1 {z + ... | :$\ds \harm 1 {n x} = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {\harm 1 {x - \dfrac k n } } + \ln n$ | {{begin-eqn}}
{{eqn | l = \map \psi {n z}
| r = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \map \psi {z + \frac k n} + \ln n
| c = [[Digamma Additive Formula]]
}}
{{eqn | ll= \leadsto
| l = \paren {-\gamma + \harm 1 {n z - 1} }
| r = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {-\gamma + \harm 1 {... | General Harmonic Number Additive Formula | https://proofwiki.org/wiki/General_Harmonic_Number_Additive_Formula | https://proofwiki.org/wiki/General_Harmonic_Number_Additive_Formula | [
"General Harmonic Number Additive Formula",
"General Harmonic Numbers"
] | [] | [
"Digamma Additive Formula",
"Digamma Function in terms of General Harmonic Number"
] |
proofwiki-21608 | Boolean Algebra as Boolean Lattice | Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra.
Then, there is a unique ordering $\preceq$ on $S$ such that:
:$\struct {S, \vee, \wedge, \preceq}$ is a Boolean lattice
In particular, it can be defined by:
:$a \preceq b \iff a \vee b = b$
for all $a, b \in S$. | First, we will show that the given ordering indeed forms a Boolean lattice.
{{Recall|Boolean Lattice|index = 2}}
{{:Definition:Boolean Lattice/Definition 2}}
{{Recall|Ordered Structure}}
{{:Definition:Ordered Structure}}
Hence it suffices to show that:
:$(1): \quad \preceq$ is in fact an ordering
:$(2): \quad \preceq$ ... | Let $\struct {S, \vee, \wedge, \neg}$ be a [[Definition:Boolean Algebra|Boolean algebra]].
Then, there is a [[Definition:Unique|unique]] [[Definition:Ordering|ordering]] $\preceq$ on $S$ such that:
:$\struct {S, \vee, \wedge, \preceq}$ is a [[Definition:Boolean Lattice|Boolean lattice]]
In particular, it can be defi... | First, we will show that the given [[Definition:Ordering|ordering]] indeed forms a [[Definition:Boolean Lattice|Boolean lattice]].
{{Recall|Boolean Lattice|index = 2}}
{{:Definition:Boolean Lattice/Definition 2}}
{{Recall|Ordered Structure}}
{{:Definition:Ordered Structure}}
Hence it suffices to show that:
:$(1):... | Boolean Algebra as Boolean Lattice | https://proofwiki.org/wiki/Boolean_Algebra_as_Boolean_Lattice | https://proofwiki.org/wiki/Boolean_Algebra_as_Boolean_Lattice | [
"Boolean Algebra is Equivalent to Boolean Lattice",
"Boolean Algebras",
"Boolean Lattices"
] | [
"Definition:Boolean Algebra",
"Definition:Unique",
"Definition:Ordering",
"Definition:Boolean Lattice"
] | [
"Definition:Ordering",
"Definition:Boolean Lattice",
"Definition:Ordering",
"Definition:Relation Compatible with Operation",
"Definition:Ordering",
"Definition:Relation Compatible with Operation",
"Definition:Relation Compatible with Operation",
"Definition:Boolean Lattice",
"Definition:Ordering",
... |
proofwiki-21609 | Geometric Congruence is Equivalence Relation | Let $S$ be the set of geometric figures.
For $F_1, F_2 \in S$, let $F_1 \cong F_2$ denote that $F_1$ is congruent to $F_2$.
Then $\cong$ is an equivalence relation on $S$. | {{tidy}}
Checking in turn each of the criteria for equivalence: | Let $S$ be the [[Definition:Set|set]] of [[Definition:Geometric Figure|geometric figures]].
For $F_1, F_2 \in S$, let $F_1 \cong F_2$ denote that $F_1$ is [[Definition:Congruence (Geometry)|congruent]] to $F_2$.
Then $\cong$ is an [[Definition:Equivalence Relation|equivalence relation]] on $S$. | {{tidy}}
Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Geometric Congruence is Equivalence Relation | https://proofwiki.org/wiki/Geometric_Congruence_is_Equivalence_Relation | https://proofwiki.org/wiki/Geometric_Congruence_is_Equivalence_Relation | [
"Congruence (Geometry)",
"Examples of Equivalence Relations"
] | [
"Definition:Set",
"Definition:Geometric Figure",
"Definition:Congruence (Geometry)",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-21610 | Nth Derivative of General Harmonic Number Order One | :$\dfrac {\d^n} {\d x^n} \harm 1 x = \paren {-1}^{n + 1} n! \paren {\map \zeta {n + 1} - \harm {n + 1} x}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\d^n} {\d x^n} \harm 1 x
| r = \dfrac {\d^n} {\d x^n} \paren {\sum_{k \mathop = 1}^\infty \paren {\frac 1 {k^1} - \frac 1 {\paren {k + x}^1} } }
| c = {{Defof|General Harmonic Numbers}}
}}
{{eqn | r = \dfrac {\d^n} {\d x^n} \paren {\sum_{k \mathop = 1}^\infty \frac 1 {k^1} }... | :$\dfrac {\d^n} {\d x^n} \harm 1 x = \paren {-1}^{n + 1} n! \paren {\map \zeta {n + 1} - \harm {n + 1} x}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\d^n} {\d x^n} \harm 1 x
| r = \dfrac {\d^n} {\d x^n} \paren {\sum_{k \mathop = 1}^\infty \paren {\frac 1 {k^1} - \frac 1 {\paren {k + x}^1} } }
| c = {{Defof|General Harmonic Numbers}}
}}
{{eqn | r = \dfrac {\d^n} {\d x^n} \paren {\sum_{k \mathop = 1}^\infty \frac 1 {k^1} }... | Nth Derivative of General Harmonic Number Order One | https://proofwiki.org/wiki/Nth_Derivative_of_General_Harmonic_Number_Order_One | https://proofwiki.org/wiki/Nth_Derivative_of_General_Harmonic_Number_Order_One | [
"Derivatives",
"General Harmonic Numbers"
] | [] | [
"Linear Combination of Convergent Series",
"Category:Derivatives",
"Category:General Harmonic Numbers"
] |
proofwiki-21611 | General Harmonic Number Reflection Formula | :$\harm r {x - 1} + \paren {-1}^r \harm r {-x} = \paren {1 + \paren {-1}^r} \map \zeta r + \dfrac {\paren {-1}^r} {\paren {r - 1}!} \map {\dfrac {\d^{r - 1} } {\d x^{r - 1} } } {\pi \map \cot {\pi x} }$ | === Lemma 1 ===
{{:General Harmonic Number Reflection Formula/Lemma 1}}{{qed|lemma}} | :$\harm r {x - 1} + \paren {-1}^r \harm r {-x} = \paren {1 + \paren {-1}^r} \map \zeta r + \dfrac {\paren {-1}^r} {\paren {r - 1}!} \map {\dfrac {\d^{r - 1} } {\d x^{r - 1} } } {\pi \map \cot {\pi x} }$ | === [[General Harmonic Number Reflection Formula/Lemma 1|Lemma 1]] ===
{{:General Harmonic Number Reflection Formula/Lemma 1}}{{qed|lemma}} | General Harmonic Number Reflection Formula | https://proofwiki.org/wiki/General_Harmonic_Number_Reflection_Formula | https://proofwiki.org/wiki/General_Harmonic_Number_Reflection_Formula | [
"General Harmonic Number Reflection Formula",
"General Harmonic Numbers",
"Reflection Formulas"
] | [] | [
"General Harmonic Number Reflection Formula/Lemma 1"
] |
proofwiki-21612 | Composite Frame Homomorphism is Frame Homomorphism | Let $L_1 = \struct{S_1, \preceq_1}$, $L_2 = \struct{S_2, \preceq_2}$ and $L_3 = \struct{S_3, \preceq_3}$ be frames.
Let $\phi_1: L_1 \to L_2$ and $\phi_2: L_2 \to L_3$ be frame homomorphisms.
Let $\phi_2 \circ \phi_1 : S_1 \to S_3$ be the composite mapping of $\phi_1$ and $\phi_2$
Then:
:$\phi_2 \circ \phi_1$ is a fr... | === $\phi_2 \circ \phi_1$ is Finite Meet Preserving ===
Let $F \subseteq S_1$ be a finite subset.
We have:
{{begin-eqn}}
{{eqn | l = \inf \paren {\phi_2 \circ \phi_1} \sqbrk F
| r = \inf \phi_2 \sqbrk {\phi_1 \sqbrk F}
| c = Image of Subset under Composite Relation with Common Codomain and Domain
}}
{{eqn |... | Let $L_1 = \struct{S_1, \preceq_1}$, $L_2 = \struct{S_2, \preceq_2}$ and $L_3 = \struct{S_3, \preceq_3}$ be [[Definition:Frame (Lattice Theory)|frames]].
Let $\phi_1: L_1 \to L_2$ and $\phi_2: L_2 \to L_3$ be [[Definition:Frame Homomorphism|frame homomorphisms]].
Let $\phi_2 \circ \phi_1 : S_1 \to S_3$ be the [[Defi... | === $\phi_2 \circ \phi_1$ is Finite Meet Preserving ===
Let $F \subseteq S_1$ be a [[Definition:Finite Set|finite]] [[Definition:Subset|subset]].
We have:
{{begin-eqn}}
{{eqn | l = \inf \paren {\phi_2 \circ \phi_1} \sqbrk F
| r = \inf \phi_2 \sqbrk {\phi_1 \sqbrk F}
| c = [[Image of Subset under Composit... | Composite Frame Homomorphism is Frame Homomorphism | https://proofwiki.org/wiki/Composite_Frame_Homomorphism_is_Frame_Homomorphism | https://proofwiki.org/wiki/Composite_Frame_Homomorphism_is_Frame_Homomorphism | [
"Frame Homomorphisms"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Frame Homomorphism",
"Definition:Composition of Mappings",
"Definition:Frame Homomorphism"
] | [
"Definition:Finite Set",
"Definition:Subset",
"Image of Subset under Composite Relation with Common Codomain and Domain",
"Definition:Finite Meet Preserving Mapping",
"Definition:Subset",
"Image of Subset under Composite Relation with Common Codomain and Domain"
] |
proofwiki-21613 | Identity Mapping is Frame Homomorphism | Let $L = \struct{S, \preceq}$ be a frame.
Let $\operatorname{id}_S$ denote the identity mapping on $S$.
Then:
:$\operatorname{id}_S$ is a frame homomorphism of $L$ to $L$ | === $\operatorname{id}_S$ is Finite Meet Preserving ===
Let $F \subseteq S_1$ be a finite subset.
We have:
{{begin-eqn}}
{{eqn | l = \inf \operatorname{id}_S \sqbrk F
| r = \inf F
| c = {{Defof|Identity Mapping}}
}}
{{eqn | r = \map {\operatorname{id}_S} {\inf F}
| c = {{Defof|Identity Mapping}}
}}
{{... | Let $L = \struct{S, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Let $\operatorname{id}_S$ denote the [[Definition:Identity Mapping|identity mapping]] on $S$.
Then:
:$\operatorname{id}_S$ is a [[Definition:Frame Homomorphism|frame homomorphism]] of $L$ to $L$ | === $\operatorname{id}_S$ is Finite Meet Preserving ===
Let $F \subseteq S_1$ be a [[Definition:Finite Set|finite]] [[Definition:Subset|subset]].
We have:
{{begin-eqn}}
{{eqn | l = \inf \operatorname{id}_S \sqbrk F
| r = \inf F
| c = {{Defof|Identity Mapping}}
}}
{{eqn | r = \map {\operatorname{id}_S} {\i... | Identity Mapping is Frame Homomorphism | https://proofwiki.org/wiki/Identity_Mapping_is_Frame_Homomorphism | https://proofwiki.org/wiki/Identity_Mapping_is_Frame_Homomorphism | [
"Frame Homomorphisms"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Identity Mapping",
"Definition:Frame Homomorphism"
] | [
"Definition:Finite Set",
"Definition:Subset",
"Definition:Finite Meet Preserving Mapping",
"Definition:Subset"
] |
proofwiki-21614 | Escape Speed in terms of Universal Gravitational Constant | Let $P$ be a planet.
Let $P$ have:
:a mass of $M$
:a radius of $R$.
Then the escape speed of $P$ is given by:
:$V = \sqrt {\dfrac {2 M G} R}$
where $G$ is the universal gravitational constant. | Let $b$ be a particle with mass $m$.
Let $K_i$ be the kinetic energy of $b$ at the surface of $P$.
Let $U_i$ be the potential energy of $b$ at the surface of $P$.
Let $K_f$ be the limit of the kinetic energy of $b$ as it leaves $P$.
Let $U_f$ be the limit of the potential energy of $b$ as it leaves $P$.
Let $v_f... | Let $P$ be a [[Definition:Planet|planet]].
Let $P$ have:
:a [[Definition:Mass|mass]] of $M$
:a [[Definition:Radius of Sphere|radius]] of $R$.
Then the [[Definition:Escape Speed|escape speed]] of $P$ is given by:
:$V = \sqrt {\dfrac {2 M G} R}$
where $G$ is the [[Definition:Universal Gravitational Constant|universal ... | Let $b$ be a [[Definition:Particle|particle]] with [[Definition:Mass|mass]] $m$.
Let $K_i$ be the [[Definition:Kinetic Energy|kinetic energy]] of $b$ at the [[Definition:Surface|surface]] of $P$.
Let $U_i$ be the [[Definition:Potential Energy|potential energy]] of $b$ at the [[Definition:Surface|surface]] of $P$.... | Escape Speed in terms of Universal Gravitational Constant | https://proofwiki.org/wiki/Escape_Speed_in_terms_of_Universal_Gravitational_Constant | https://proofwiki.org/wiki/Escape_Speed_in_terms_of_Universal_Gravitational_Constant | [
"Escape Speed"
] | [
"Definition:Planet",
"Definition:Mass",
"Definition:Sphere/Geometry/Radius",
"Definition:Escape Speed",
"Definition:Universal Gravitational Constant"
] | [
"Definition:Particle",
"Definition:Mass",
"Definition:Kinetic Energy",
"Definition:Surface",
"Definition:Potential Energy",
"Definition:Surface",
"Definition:Limit",
"Definition:Potential Energy",
"Definition:Speed",
"Definition:Distance",
"Definition:Center",
"Definition:Surface",
"Definiti... |
proofwiki-21615 | Characterization of Compact Element in Complete Lattice | Let $L = \struct{S, \preceq}$ be a complete lattice.
Let $a \in S$.
{{TFAE}}:
:$(1)\quad a$ is a compact element
:$(2)\quad \forall I \subseteq S : I$ is an ideal $: a \preceq \sup I \implies a \in I$
:$(3)\quad \forall A \subseteq S : a \preceq \sup A \implies \exists F \subseteq A : F$ is finite $: a \preceq \sup F$ | Recall, $a$ is a compact element {{iff}}:
:for every directed subset $D$ of $S$ such that $a \preceq \sup D$
::$\exists d \in D: a \preceq d$ | Let $L = \struct{S, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $a \in S$.
{{TFAE}}:
:$(1)\quad a$ is a [[Definition:Compact Element|compact element]]
:$(2)\quad \forall I \subseteq S : I$ is an [[Definition:Lattice Ideal|ideal]] $: a \preceq \sup I \implies a \in I$
:$(3)\quad \forall A \... | Recall, $a$ is a [[Definition:Compact Element|compact element]] {{iff}}:
:for every [[Definition:Directed Subset|directed subset]] $D$ of $S$ such that $a \preceq \sup D$
::$\exists d \in D: a \preceq d$ | Characterization of Compact Element in Complete Lattice | https://proofwiki.org/wiki/Characterization_of_Compact_Element_in_Complete_Lattice | https://proofwiki.org/wiki/Characterization_of_Compact_Element_in_Complete_Lattice | [
"Complete Lattices",
"Characterization of Compact Element in Complete Lattice"
] | [
"Definition:Complete Lattice",
"Definition:Compact Element",
"Definition:Lattice Ideal",
"Definition:Finite Set"
] | [
"Definition:Compact Element",
"Definition:Directed Subset",
"Definition:Compact Element"
] |
proofwiki-21616 | Characterization of Compact Element in Frame or Locale | Let $L = \struct{S, \preceq}$ be a frame or locale.
Let $a \in S$.
{{TFAE}}:
:$(1)\quad a$ is a compact element
:$(2)\quad \forall I \subseteq S : I$ is an ideal $: a \preceq \sup I \implies a \in I$
:$(3)\quad \forall A \subseteq S : a \preceq \sup A \implies \exists F \subseteq A : F$ is finite $: a \preceq \sup F$
... | Recall, a frame or locale is a complete lattice satisfying the infinite join distributive law:
{{:Axiom:Infinite Join Distributive Law}}
From Characterization of Compact Element in Complete Lattice:
:Statements $(1)$, $(2)$ and $(3)$ are equivalent. | Let $L = \struct{S, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]] or [[Definition:Locale (Lattice Theory)|locale]].
Let $a \in S$.
{{TFAE}}:
:$(1)\quad a$ is a [[Definition:Compact Element|compact element]]
:$(2)\quad \forall I \subseteq S : I$ is an [[Definition:Lattice Ideal|ideal]] $: a \preceq \sup... | Recall, a [[Definition:Frame (Lattice Theory)|frame]] or [[Definition:Locale (Lattice Theory)|locale]] is a [[Definition:Complete Lattice|complete lattice]] satisfying the [[Axiom:Infinite Join Distributive Law|infinite join distributive law]]:
{{:Axiom:Infinite Join Distributive Law}}
From [[Characterization of Comp... | Characterization of Compact Element in Frame or Locale | https://proofwiki.org/wiki/Characterization_of_Compact_Element_in_Frame_or_Locale | https://proofwiki.org/wiki/Characterization_of_Compact_Element_in_Frame_or_Locale | [
"Frames",
"Locales"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Locale (Lattice Theory)",
"Definition:Compact Element",
"Definition:Lattice Ideal",
"Definition:Finite Set",
"Definition:Finite Set"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Locale (Lattice Theory)",
"Definition:Complete Lattice",
"Axiom:Infinite Join Distributive Law",
"Characterization of Compact Element in Complete Lattice",
"Axiom:Infinite Join Distributive Law",
"Axiom:Infinite Join Distributive Law"
] |
proofwiki-21617 | Reciprocal of Difference of Squares as Sum of Reciprocals | :$\dfrac 1 {x^2 - y^2} = \dfrac 1 {2 x \paren {x + y} } + \dfrac 1 {2 x \paren {x - y} }$ | {{begin-eqn}}
{{eqn | l = \dfrac 1 {x + y} + \dfrac 1 {x - y}
| r = \dfrac {\paren {x - y} + \paren {x + y} } {\paren {x - y} \paren {x + y} }
| c = putting everything over a common denominator
}}
{{eqn | r = \dfrac {2 x} {x^2 - y^2}
| c = simplifying, and Difference of Two Squares
}}
{{eqn | ll= \lea... | :$\dfrac 1 {x^2 - y^2} = \dfrac 1 {2 x \paren {x + y} } + \dfrac 1 {2 x \paren {x - y} }$ | {{begin-eqn}}
{{eqn | l = \dfrac 1 {x + y} + \dfrac 1 {x - y}
| r = \dfrac {\paren {x - y} + \paren {x + y} } {\paren {x - y} \paren {x + y} }
| c = putting everything over a [[Definition:Common Denominator|common denominator]]
}}
{{eqn | r = \dfrac {2 x} {x^2 - y^2}
| c = simplifying, and [[Differenc... | Reciprocal of Difference of Squares as Sum of Reciprocals | https://proofwiki.org/wiki/Reciprocal_of_Difference_of_Squares_as_Sum_of_Reciprocals | https://proofwiki.org/wiki/Reciprocal_of_Difference_of_Squares_as_Sum_of_Reciprocals | [
"Reciprocal of Difference of Squares",
"Algebra"
] | [] | [
"Definition:Common Denominator",
"Difference of Two Squares"
] |
proofwiki-21618 | Reciprocal of Difference of Squares as Difference of Reciprocals | :$\dfrac 1 {x^2 - y^2} = \dfrac 1 {2 y \paren {x - y} } - \dfrac 1 {2 y \paren {x + y} }$ | {{begin-eqn}}
{{eqn | l = \dfrac 1 {x - y} - \dfrac 1 {x + y}
| r = \dfrac {\paren {x + y} - \paren {x - y} } {\paren {x + y} \paren {x - y} }
| c = putting everything over a common denominator
}}
{{eqn | r = \dfrac {2 y} {x^2 - y^2}
| c = simplifying, and Difference of Two Squares
}}
{{eqn | ll= \lea... | :$\dfrac 1 {x^2 - y^2} = \dfrac 1 {2 y \paren {x - y} } - \dfrac 1 {2 y \paren {x + y} }$ | {{begin-eqn}}
{{eqn | l = \dfrac 1 {x - y} - \dfrac 1 {x + y}
| r = \dfrac {\paren {x + y} - \paren {x - y} } {\paren {x + y} \paren {x - y} }
| c = putting everything over a [[Definition:Common Denominator|common denominator]]
}}
{{eqn | r = \dfrac {2 y} {x^2 - y^2}
| c = simplifying, and [[Differenc... | Reciprocal of Difference of Squares as Difference of Reciprocals | https://proofwiki.org/wiki/Reciprocal_of_Difference_of_Squares_as_Difference_of_Reciprocals | https://proofwiki.org/wiki/Reciprocal_of_Difference_of_Squares_as_Difference_of_Reciprocals | [
"Reciprocal of Difference of Squares",
"Algebra"
] | [] | [
"Definition:Common Denominator",
"Difference of Two Squares"
] |
proofwiki-21619 | Primitive of Root of x squared plus a squared over x/Logarithm Form | :$\ds \int \frac {\sqrt {x^2 + a^2} } x \rd x = \sqrt {x^2 + a^2} - a \map \ln {\frac {a + \sqrt {x^2 + a^2} } a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 + a^2} } x \rd x
| r = \int \frac {\sqrt {z + a^2} \rd z} {2 \sqrt z \sqrt z}
| c = Integ... | :$\ds \int \frac {\sqrt {x^2 + a^2} } x \rd x = \sqrt {x^2 + a^2} - a \map \ln {\frac {a + \sqrt {x^2 + a^2} } a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 + a^2} } x \rd x
| r = \int \frac {\sqrt {z + a^2} \rd z} {2 \sqrt z \sqrt z}
| c = [... | Primitive of Root of x squared plus a squared over x/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_over_x/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_over_x/Logarithm_Form | [
"Primitive of Root of x squared plus a squared over x"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b over x",
"Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form"
] |
proofwiki-21620 | Primitive of Root of x squared plus a squared over x squared/Logarithm Form | :$\ds \int \frac {\sqrt {x^2 + a^2} } {x^2} \rd x = \frac {-\sqrt {x^2 + a^2} } x + \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 + a^2} } {x^2} \rd x
| r = \int \frac {\sqrt {z + a^2} \rd z} {2 z \sqrt z}
| c = Integra... | :$\ds \int \frac {\sqrt {x^2 + a^2} } {x^2} \rd x = \frac {-\sqrt {x^2 + a^2} } x + \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 + a^2} } {x^2} \rd x
| r = \int \frac {\sqrt {z + a^2} \rd z} {2 z \sqrt z}
| c = [[I... | Primitive of Root of x squared plus a squared over x squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_over_x_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_over_x_squared/Logarithm_Form | [
"Primitive of Root of x squared plus a squared over x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b over Power of x/Formulation 1",
"Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form"
] |
proofwiki-21621 | Row Rank of Matrix equals Rank of Matrix | Let $\mathbf A$ be a matrix.
The row rank of $\mathbf A$ is equal to the rank of $\mathbf A$. | The rank of $\mathbf A$ is defined as the dimension of the column space of $\mathbf A$.
That is, the rank of $\mathbf A$ is the column rank of $\mathbf A$.
The result follows from Column Rank of Matrix equals Row Rank.
{{qed}} | Let $\mathbf A$ be a [[Definition:Matrix|matrix]].
The [[Definition:Row Rank|row rank]] of $\mathbf A$ is equal to the [[Definition:Rank of Matrix|rank]] of $\mathbf A$. | The [[Definition:Rank of Matrix|rank]] of $\mathbf A$ is defined as the [[Definition:Dimension of Vector Space|dimension]] of the [[Definition:Column Space|column space]] of $\mathbf A$.
That is, the [[Definition:Rank of Matrix|rank]] of $\mathbf A$ is the [[Definition:Column Rank|column rank]] of $\mathbf A$.
The re... | Row Rank of Matrix equals Rank of Matrix | https://proofwiki.org/wiki/Row_Rank_of_Matrix_equals_Rank_of_Matrix | https://proofwiki.org/wiki/Row_Rank_of_Matrix_equals_Rank_of_Matrix | [
"Row Rank",
"Rank of Matrix"
] | [
"Definition:Matrix",
"Definition:Row Rank",
"Definition:Rank/Matrix"
] | [
"Definition:Rank/Matrix",
"Definition:Dimension of Vector Space",
"Definition:Column Space",
"Definition:Rank/Matrix",
"Definition:Column Rank",
"Column Rank of Matrix equals Row Rank"
] |
proofwiki-21622 | Primitive of Power of Root of x squared minus a squared | :$\ds \int \paren {\sqrt {x^2 - a^2} }^n \rd x = \dfrac {x \paren {\sqrt {x^2 - a^2} }^n} {n + 1} - \dfrac {n a^2} {n + 1} \int \paren {\sqrt {x^2 - a^2} }^{n - 2} \rd x$
for $n \ne -1$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sinh \theta
| c = Derivative of Hyperbolic Cosine
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = x^... | :$\ds \int \paren {\sqrt {x^2 - a^2} }^n \rd x = \dfrac {x \paren {\sqrt {x^2 - a^2} }^n} {n + 1} - \dfrac {n a^2} {n + 1} \int \paren {\sqrt {x^2 - a^2} }^{n - 2} \rd x$
for $n \ne -1$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sinh \theta
| c = [[Derivative of Hyperbolic Cosine]]
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| ... | Primitive of Power of Root of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_x_squared_minus_a_squared | [
"Primitive of Power of Root of x squared minus a squared",
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power of Hyperbolic Sine of a x"
] |
proofwiki-21623 | Primitive of Reciprocal of Power of Root of x squared minus a squared | :$\ds \int \dfrac {\d x} {\paren {\sqrt {x^2 - a^2} }^n} = \dfrac {x \paren {\sqrt {x^2 - a^2} }^{2 - n} } {\paren {2 - n} a^2} - \dfrac {n - 3} {\paren {n - 2} a^2} \int \dfrac {\d x} {\paren {\sqrt {x^2 - a^2} }^{n - 2} }$
for $n \ne 2$. | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sinh \theta
| c = Derivative of Hyperbolic Cosine
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = x^... | :$\ds \int \dfrac {\d x} {\paren {\sqrt {x^2 - a^2} }^n} = \dfrac {x \paren {\sqrt {x^2 - a^2} }^{2 - n} } {\paren {2 - n} a^2} - \dfrac {n - 3} {\paren {n - 2} a^2} \int \dfrac {\d x} {\paren {\sqrt {x^2 - a^2} }^{n - 2} }$
for $n \ne 2$. | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sinh \theta
| c = [[Derivative of Hyperbolic Cosine]]
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| ... | Primitive of Reciprocal of Power of Root of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Root_of_x_squared_minus_a_squared | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of Power of Hyperbolic Sine of a x"
] |
proofwiki-21624 | Primitive of x by Power of Root of x squared minus a squared | :$\ds \int x \paren {\sqrt {x^2 - a^2} }^n \rd x = \dfrac {\paren {\sqrt {x^2 - a^2} }^{n + 2} } {n + 2} + C$
for $n \ne -2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Derivative of Power
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d z} 2
| r = x \rd x
| c =
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l = \int x \paren {\sqrt {... | :$\ds \int x \paren {\sqrt {x^2 - a^2} }^n \rd x = \dfrac {\paren {\sqrt {x^2 - a^2} }^{n + 2} } {n + 2} + C$
for $n \ne -2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Derivative of Power]]
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d z} 2
| r = x \rd x
| c =
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l = \int x \paren {\... | Primitive of x by Power of Root of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_x_by_Power_of_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_by_Power_of_Root_of_x_squared_minus_a_squared | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power of Root of a x + b"
] |
proofwiki-21625 | Primitive of Root of x squared minus a squared over x squared/Logarithm Form | :$\ds \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x = \frac {-\sqrt {x^2 - a^2} } x + \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $x^2 \ge a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x
| r = \int \frac {\sqrt {z - a^2} \rd z} {2 z \sqrt z}
| c = Integra... | :$\ds \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x = \frac {-\sqrt {x^2 - a^2} } x + \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $x^2 \ge a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x
| r = \int \frac {\sqrt {z - a^2} \rd z} {2 z \sqrt z}
| c = [[I... | Primitive of Root of x squared minus a squared over x squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_over_x_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_over_x_squared/Logarithm_Form | [
"Primitive of Root of x squared minus a squared over x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b over Power of x/Formulation 1",
"Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form"
] |
proofwiki-21626 | Primitive of Root of x squared minus a squared over x squared/Inverse Hyperbolic Cosine Form | :$\ds \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x = \arcosh \dfrac x a - \frac {\sqrt {x^2 - a^2} } x + C$
for $x^2 \ge a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x
| r = \frac {-\sqrt {x^2 - a^2} } x + \ln \size {x + \sqrt {x^2 - a^2} } + C
| c = Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$
}}
{{eqn | r = \frac {-\sqrt {x^2 - a^2} } x + \map \ln {x + \sqrt {x^2 - a^2} } + C
| c = as $x + ... | :$\ds \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x = \arcosh \dfrac x a - \frac {\sqrt {x^2 - a^2} } x + C$
for $x^2 \ge a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x
| r = \frac {-\sqrt {x^2 - a^2} } x + \ln \size {x + \sqrt {x^2 - a^2} } + C
| c = [[Primitive of Root of x squared minus a squared over x squared/Logarithm Form|Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$]]
}}
{{eqn | r = \frac {-\... | Primitive of Root of x squared minus a squared over x squared/Inverse Hyperbolic Cosine Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_over_x_squared/Inverse_Hyperbolic_Cosine_Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_over_x_squared/Inverse_Hyperbolic_Cosine_Form | [
"Primitive of Root of x squared minus a squared over x squared"
] | [] | [
"Primitive of Root of x squared minus a squared over x squared/Logarithm Form",
"Definition:Primitive (Calculus)/Constant of Integration",
"Real Area Hyperbolic Cosine of x over a in Logarithm Form"
] |
proofwiki-21627 | Primitive of Power of Root of a squared minus x squared | :$\ds \int \paren {\sqrt {a^2 - x^2} }^n \rd x = \dfrac {x \paren {\sqrt {a^2 - x^2} }^n} {n + 1} - \dfrac {n a^2} {n + 1} \int \paren {\sqrt {a^2 - x^2} }^{n - 2} \rd x$
for $n \ne -1$. | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \cos \theta
| c = Derivative of Sine Function
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin \theta
}}
{{eqn | ll= \leadsto
| l = \sqrt {a^... | :$\ds \int \paren {\sqrt {a^2 - x^2} }^n \rd x = \dfrac {x \paren {\sqrt {a^2 - x^2} }^n} {n + 1} - \dfrac {n a^2} {n + 1} \int \paren {\sqrt {a^2 - x^2} }^{n - 2} \rd x$
for $n \ne -1$. | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \cos \theta
| c = [[Derivative of Sine Function]]
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin \theta
}}
{{eqn | ll= \leadsto
| l = \s... | Primitive of Power of Root of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_a_squared_minus_x_squared | [
"Primitive of Power of Root of a squared minus x squared",
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Derivative of Sine Function",
"Sum of Squares of Sine and Cosine",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Reduction Formula for Integral of Power of Cosine/Corollary"
] |
proofwiki-21628 | Primitive of Power of Root of 2 a x minus x squared | :$\ds \int \paren {\sqrt {2 a x - x^2} }^n \rd x = \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^n} {n + 1} + \frac {n a^2} {n + 1} \int \paren {\sqrt {2 a x - x^2} }^{n - 2} \rd x$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
{{begin-eqn}}
{{eqn | l = 2 a x - x^2
| r = 2 a \paren {u + a} - \paren {u + a}^2
| c =
}}
{{eqn | r = 2 a u + 2 a^2 - u^2 - 2 a u - a^2
| c =
}}
{{eqn | r = a^2 - u^2
| c =
}}
{{end-eqn}}
and we have:
{{begin-eqn}}
{... | :$\ds \int \paren {\sqrt {2 a x - x^2} }^n \rd x = \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^n} {n + 1} + \frac {n a^2} {n + 1} \int \paren {\sqrt {2 a x - x^2} }^{n - 2} \rd x$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
{{begin-eqn}}
{{eqn | l = 2 a x - x^2
| r = 2 a \paren {u + a} - \paren {u + a}^2
| c =
}}
{{eqn | r = 2 a u + 2 a^2 - u^2 - 2 a u - a^2
| c =
}}
{{eqn | r = a^2 - u^2
| c =
}}
{{end-eqn}}
and we have:
{{begin-... | Primitive of Power of Root of 2 a x minus x squared | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_2_a_x_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_2_a_x_minus_x_squared | [
"Primitives of Roots of Quadratic Functions"
] | [] | [
"Primitive of Power of Root of a squared minus x squared"
] |
proofwiki-21629 | Primitive of Reciprocal of Power of Root of 2 a x minus x squared | :$\ds \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^n} = \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^{2 - n} } {\paren {n - 2} a^2} + \frac {n - 3} {\paren {n - 2} a^2} \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^{n - 2} }$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^{n - 2} }
| r = \int \paren {\sqrt {2 a x - x^2} }^{2 - n} \rd x
| c =
}}
{{eqn | r = \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^{2 - n} } {\paren {2 - n} + 1} + \frac {\paren {2 - n} a^2} {\paren {2 - n} + 1} \int \paren ... | :$\ds \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^n} = \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^{2 - n} } {\paren {n - 2} a^2} + \frac {n - 3} {\paren {n - 2} a^2} \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^{n - 2} }$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {\d x} {\paren {\sqrt {2 a x - x^2} }^{n - 2} }
| r = \int \paren {\sqrt {2 a x - x^2} }^{2 - n} \rd x
| c =
}}
{{eqn | r = \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^{2 - n} } {\paren {2 - n} + 1} + \frac {\paren {2 - n} a^2} {\paren {2 - n} + 1} \int \paren ... | Primitive of Reciprocal of Power of Root of 2 a x minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Root_of_2_a_x_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_Root_of_2_a_x_minus_x_squared | [
"Primitives of Roots of Quadratic Functions"
] | [] | [
"Primitive of Power of Root of 2 a x minus x squared",
"Definition:Multiplication/Real Numbers"
] |
proofwiki-21630 | Primitive of x by Root of 2 a x minus x squared | :$\ds \int x \sqrt {2 a x - x^2} \rd x = \frac {\paren {x + a} \paren {2 x - 3 a} \sqrt {2 a x - x^2} } 6 + \frac {a^3} 2 \arcsin \dfrac {x - a} a + C$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
{{begin-eqn}}
{{eqn | l = 2 a x - x^2
| r = 2 a \paren {u + a} - \paren {u + a}^2
| c =
}}
{{eqn | r = 2 a u + 2 a^2 - u^2 - 2 a u - a^2
| c =
}}
{{eqn | r = a^2 - u^2
| c =
}}
{{end-eqn}}
and we have:
{{begin-eqn}}
{... | :$\ds \int x \sqrt {2 a x - x^2} \rd x = \frac {\paren {x + a} \paren {2 x - 3 a} \sqrt {2 a x - x^2} } 6 + \frac {a^3} 2 \arcsin \dfrac {x - a} a + C$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
{{begin-eqn}}
{{eqn | l = 2 a x - x^2
| r = 2 a \paren {u + a} - \paren {u + a}^2
| c =
}}
{{eqn | r = 2 a u + 2 a^2 - u^2 - 2 a u - a^2
| c =
}}
{{eqn | r = a^2 - u^2
| c =
}}
{{end-eqn}}
and we have:
{{begin-... | Primitive of x by Root of 2 a x minus x squared | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_2_a_x_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_2_a_x_minus_x_squared | [
"Primitives of Roots of Quadratic Functions"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x by Root of a squared minus x squared",
"Primitive of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-21631 | Primitive of Root of 2 a x minus x squared over x | :$\ds \int \dfrac {\sqrt {2 a x - x^2} } x \rd x = \sqrt {2 a x - x^2} + a \arcsin \dfrac {x - a} a + C$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
{{begin-eqn}}
{{eqn | l = 2 a x - x^2
| r = 2 a \paren {u + a} - \paren {u + a}^2
| c =
}}
{{eqn | r = 2 a u + 2 a^2 - u^2 - 2 a u - a^2
| c =
}}
{{eqn | r = a^2 - u^2
| c =
}}
{{end-eqn}}
and we have:
{{begin-eqn}}
{... | :$\ds \int \dfrac {\sqrt {2 a x - x^2} } x \rd x = \sqrt {2 a x - x^2} + a \arcsin \dfrac {x - a} a + C$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
{{begin-eqn}}
{{eqn | l = 2 a x - x^2
| r = 2 a \paren {u + a} - \paren {u + a}^2
| c =
}}
{{eqn | r = 2 a u + 2 a^2 - u^2 - 2 a u - a^2
| c =
}}
{{eqn | r = a^2 - u^2
| c =
}}
{{end-eqn}}
and we have:
{{begin-... | Primitive of Root of 2 a x minus x squared over x | https://proofwiki.org/wiki/Primitive_of_Root_of_2_a_x_minus_x_squared_over_x | https://proofwiki.org/wiki/Primitive_of_Root_of_2_a_x_minus_x_squared_over_x | [
"Primitives of Roots of Quadratic Functions"
] | [] | [
"Difference of Two Squares"
] |
proofwiki-21632 | Primitive of Root of 2 a x minus x squared over x squared | :$\ds \int \dfrac {\sqrt {2 a x - x^2} } {x^2} \rd x = -2 \sqrt {\dfrac {2 a - x} x} - \arcsin \dfrac {x - a} a + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {2 a x - x^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \dfrac {2 a - 2 x} {2 \sqrt {2 a x - x^2... | :$\ds \int \dfrac {\sqrt {2 a x - x^2} } {x^2} \rd x = -2 \sqrt {\dfrac {2 a - x} x} - \arcsin \dfrac {x - a} a + C$ | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {2 a x - x^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \df... | Primitive of Root of 2 a x minus x squared over x squared | https://proofwiki.org/wiki/Primitive_of_Root_of_2_a_x_minus_x_squared_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Root_of_2_a_x_minus_x_squared_over_x_squared | [
"Primitives of Roots of Quadratic Functions"
] | [] | [
"Definition:Primitive (Calculus)",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x by Root of 2 a x minus x squared",
"Primitive of Reciprocal of Root of ... |
proofwiki-21633 | Characterization of Locale | Let $L = \struct{S, \preceq}$ be an ordered set.
{{TFAE}}
:$(1): \quad L$ is a locale
:$(2): \quad L$ is a frame
:$(3): \quad L$ is a complete lattice satisfying the infinite join distributive law
:$(4): \quad L$ is a complete Heyting algebra
:$(5): \quad L$ is a complete Brouwerian lattice | === Statement $(1)$ {{iff}} Statement $(2)$ ===
The equivalence of Statement $(1)$ and Statement $(2)$ follows immediately from the definition of locale
{{qed|lemma}} | Let $L = \struct{S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
{{TFAE}}
:$(1): \quad L$ is a [[Definition:Locale (Lattice Theory)|locale]]
:$(2): \quad L$ is a [[Definition:Frame (Lattice Theory)|frame]]
:$(3): \quad L$ is a [[Definition:Complete Lattice|complete lattice]] satisfying the [[Axiom:Infinite... | === Statement $(1)$ {{iff}} Statement $(2)$ ===
The [[Definition:Equivalent Statements|equivalence]] of Statement $(1)$ and Statement $(2)$ follows immediately from the definition of [[Definition:Locale (Lattice Theory)|locale]]
{{qed|lemma}} | Characterization of Locale | https://proofwiki.org/wiki/Characterization_of_Locale | https://proofwiki.org/wiki/Characterization_of_Locale | [
"Locales",
"Frames",
"Brouwerian Lattices",
"Heyting Algebras",
"Complete Lattices",
"Characterization of Locale"
] | [
"Definition:Ordered Set",
"Definition:Locale (Lattice Theory)",
"Definition:Frame (Lattice Theory)",
"Definition:Complete Lattice",
"Axiom:Infinite Join Distributive Law",
"Definition:Complete Lattice",
"Definition:Heyting Algebra",
"Definition:Complete Lattice",
"Definition:Brouwerian Lattice"
] | [
"Definition:Logical Equivalence",
"Definition:Locale (Lattice Theory)",
"Definition:Logical Equivalence"
] |
proofwiki-21634 | Primitive of x over Root of 2 a x minus x squared | :$\ds \int \dfrac x {\sqrt {2 a x - x^2} } \rd x = a \arcsin \dfrac {x - a} a - \sqrt {2 a x - x^2} + C$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
{{begin-eqn}}
{{eqn | l = 2 a x - x^2
| r = 2 a \paren {u + a} - \paren {u + a}^2
| c =
}}
{{eqn | r = 2 a u + 2 a^2 - u^2 - 2 a u - a^2
| c =
}}
{{eqn | r = a^2 - u^2
| c =
}}
{{end-eqn}}
and we have:
{{begin-eqn}}
{... | :$\ds \int \dfrac x {\sqrt {2 a x - x^2} } \rd x = a \arcsin \dfrac {x - a} a - \sqrt {2 a x - x^2} + C$ | Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
{{begin-eqn}}
{{eqn | l = 2 a x - x^2
| r = 2 a \paren {u + a} - \paren {u + a}^2
| c =
}}
{{eqn | r = 2 a u + 2 a^2 - u^2 - 2 a u - a^2
| c =
}}
{{eqn | r = a^2 - u^2
| c =
}}
{{end-eqn}}
and we have:
{{begin-... | Primitive of x over Root of 2 a x minus x squared | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_2_a_x_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_2_a_x_minus_x_squared | [
"Primitives of Roots of Quadratic Functions"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x over Root of a squared minus x squared",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-21635 | Primitive of Reciprocal of x by Root of 2 a x minus x squared | :$\ds \int \dfrac 1 {x \sqrt {2 a x - x^2} } \rd x = -\dfrac 1 a \sqrt {\dfrac {2 a - x} x} + C$ | Recall Primitive of $\dfrac 1 {\paren {p x + q} \sqrt {\paren {a x + b} \paren {p x + q} } }$:
{{:Primitive of Reciprocal of p x + q by Root of a x + b by Root of p x + q}}
Then:
{{begin-eqn}}
{{eqn | l = \int \dfrac 1 {x \sqrt {2 a x - x^2} } \rd x
| r = \int \dfrac 1 {x \sqrt {x \paren {2 a - x} } } \rd x
... | :$\ds \int \dfrac 1 {x \sqrt {2 a x - x^2} } \rd x = -\dfrac 1 a \sqrt {\dfrac {2 a - x} x} + C$ | Recall [[Primitive of Reciprocal of p x + q by Root of a x + b by Root of p x + q|Primitive of $\dfrac 1 {\paren {p x + q} \sqrt {\paren {a x + b} \paren {p x + q} } }$]]:
{{:Primitive of Reciprocal of p x + q by Root of a x + b by Root of p x + q}}
Then:
{{begin-eqn}}
{{eqn | l = \int \dfrac 1 {x \sqrt {2 a x - x^2}... | Primitive of Reciprocal of x by Root of 2 a x minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_2_a_x_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_2_a_x_minus_x_squared | [
"Primitives of Roots of Quadratic Functions"
] | [] | [
"Primitive of Reciprocal of p x + q by Root of a x + b by Root of p x + q"
] |
proofwiki-21636 | Primitive of Cosine of a x over Sine of a x | :$\ds \int \dfrac {\cos a x} {\sin a x} \rd x = \dfrac 1 a \ln \size {\sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {\cos a x} {\sin a x} \rd x
| r = \int \cot a x \rd x
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \dfrac 1 a \ln \size {\sin a x} + C
| c = Primitive of $\cot a x$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \dfrac {\cos a x} {\sin a x} \rd x = \dfrac 1 a \ln \size {\sin a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {\cos a x} {\sin a x} \rd x
| r = \int \cot a x \rd x
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \dfrac 1 a \ln \size {\sin a x} + C
| c = [[Primitive of Cotangent of a x|Primitive of $\cot a x$]]
}}
{{end-eqn}}
{{qed}} | Primitive of Cosine of a x over Sine of a x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Sine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Sine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Cotangent is Cosine divided by Sine",
"Primitive of Cotangent of a x"
] |
proofwiki-21637 | Primitive of Sine of a x over Cosine of a x | :$\ds \int \dfrac {\sin a x} {\cos a x} \rd x = -\dfrac 1 a \ln \size {\cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {\sin a x} {\cos a x} \rd x
| r = \int \cot a x \rd x
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = -\dfrac 1 a \ln \size {\cos a x} + C
| c = Primitive of $\tan a x$: Cosine Form
}}
{{end-eqn}}
{{qed}} | :$\ds \int \dfrac {\sin a x} {\cos a x} \rd x = -\dfrac 1 a \ln \size {\cos a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {\sin a x} {\cos a x} \rd x
| r = \int \cot a x \rd x
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = -\dfrac 1 a \ln \size {\cos a x} + C
| c = [[Primitive of Tangent of a x/Cosine Form|Primitive of $\tan a x$: Cosine Form]]
}}
{{end-eqn}}
{{qed}} | Primitive of Sine of a x over Cosine of a x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Cosine_of_a_x | [
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Primitive of Tangent of a x/Cosine Form"
] |
proofwiki-21638 | Primitive of Power of x by Sine of a x/Lemma | :$\ds \int x^m \sin a x \rd x = \frac {- x^m \cos a x} a + \frac m a \int x^{m - 1} \cos a x \rd x$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^m
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = m x^{m - 1}
| c = Power Rule for Derivatives
}}
{{... | :$\ds \int x^m \sin a x \rd x = \frac {- x^m \cos a x} a + \frac m a \int x^{m - 1} \cos a x \rd x$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^m
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = m x^{m - 1}
| c = [[Powe... | Primitive of Power of x by Sine of a x/Lemma | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Sine_of_a_x/Lemma | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Sine_of_a_x/Lemma | [
"Primitive of Power of x by Sine of a x"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Sine Function/Corollary",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21639 | Primitive of Power of x by Cosine of a x/Lemma | :$\ds \int x^m \cos a x \rd x = \frac {x^m \sin a x} a - \frac m a \int x^{m - 1} \sin a x \rd x$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^m
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = m x^{m - 1}
| c = Power Rule for Derivatives
}}
{{... | :$\ds \int x^m \cos a x \rd x = \frac {x^m \sin a x} a - \frac m a \int x^{m - 1} \sin a x \rd x$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^m
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = m x^{m - 1}
| c = [[Powe... | Primitive of Power of x by Cosine of a x/Lemma | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Cosine_of_a_x/Lemma | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Cosine_of_a_x/Lemma | [
"Primitive of Power of x by Cosine of a x",
"Primitives involving Cosine Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Cosine Function/Corollary",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21640 | Primitive of Cosecant of a x/Cosecant plus Cotangent Form | :$\ds \int \csc a x \rd x = -\frac 1 a \ln \size {\csc a x + \cot a x} + C$
where $\csc a x + \cot a x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csc x \rd x
| r = -\ln \size {\csc x + \cot x} + C
| c = Primitive of $\csc x$: Cosecant plus Cotangent Form
}}
{{eqn | ll= \leadsto
| l = \int \csc a x \rd x
| r = -\frac 1 a \ln \size {\csc a x + \cot a x} + C
| c = Primitive of Function of Constant Multip... | :$\ds \int \csc a x \rd x = -\frac 1 a \ln \size {\csc a x + \cot a x} + C$
where $\csc a x + \cot a x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \csc x \rd x
| r = -\ln \size {\csc x + \cot x} + C
| c = [[Primitive of Cosecant Function/Cosecant plus Cotangent Form|Primitive of $\csc x$: Cosecant plus Cotangent Form]]
}}
{{eqn | ll= \leadsto
| l = \int \csc a x \rd x
| r = -\frac 1 a \ln \size {\csc a x + \c... | Primitive of Cosecant of a x/Cosecant plus Cotangent Form | https://proofwiki.org/wiki/Primitive_of_Cosecant_of_a_x/Cosecant_plus_Cotangent_Form | https://proofwiki.org/wiki/Primitive_of_Cosecant_of_a_x/Cosecant_plus_Cotangent_Form | [
"Primitive of Cosecant of a x"
] | [] | [
"Primitive of Cosecant Function/Cosecant plus Cotangent Form",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21641 | Primitive of Arcsine of a x | :$\ds \int \arcsin a x \rd x = x \arcsin a x + \dfrac 1 a \sqrt {1 - a^2 x^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arcsin x \rd x
| r = x \arcsin x + \sqrt {1 - x^2} + C
| c = Primitive of $\arcsin x$
}}
{{eqn | ll= \leadsto
| l = \int \arcsin a x \rd x
| r = \dfrac 1 a \paren {\paren {a x} \arcsin a x + \sqrt {1 - \paren {a x}^2} } + C
| c = Primitive of Function of Con... | :$\ds \int \arcsin a x \rd x = x \arcsin a x + \dfrac 1 a \sqrt {1 - a^2 x^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arcsin x \rd x
| r = x \arcsin x + \sqrt {1 - x^2} + C
| c = [[Primitive of Arcsine Function|Primitive of $\arcsin x$]]
}}
{{eqn | ll= \leadsto
| l = \int \arcsin a x \rd x
| r = \dfrac 1 a \paren {\paren {a x} \arcsin a x + \sqrt {1 - \paren {a x}^2} } + C
... | Primitive of Arcsine of a x | https://proofwiki.org/wiki/Primitive_of_Arcsine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Arcsine_of_a_x | [
"Primitives involving Inverse Sine Function"
] | [] | [
"Primitive of Arcsine Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21642 | Primitive of Arctangent of a x | :$\ds \int \arctan a x \rd x = x \arctan a x - \dfrac 1 {2 a} \map \ln {1 + a^2 x^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arctan x \rd x
| r = x \arctan x - \dfrac 1 2 \map \ln {1 + x^2} + C
| c = Primitive of $\arctan x$
}}
{{eqn | ll= \leadsto
| l = \int \arctan a x \rd x
| r = \dfrac 1 a \paren {\paren {a x} \arctan a x - \dfrac 1 2 \map \ln {1 + a^2 x^2} } + C
| c = Primiti... | :$\ds \int \arctan a x \rd x = x \arctan a x - \dfrac 1 {2 a} \map \ln {1 + a^2 x^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \arctan x \rd x
| r = x \arctan x - \dfrac 1 2 \map \ln {1 + x^2} + C
| c = [[Primitive of Arctangent Function|Primitive of $\arctan x$]]
}}
{{eqn | ll= \leadsto
| l = \int \arctan a x \rd x
| r = \dfrac 1 a \paren {\paren {a x} \arctan a x - \dfrac 1 2 \map \ln {1... | Primitive of Arctangent of a x | https://proofwiki.org/wiki/Primitive_of_Arctangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Arctangent_of_a_x | [
"Primitives involving Inverse Tangent Function"
] | [] | [
"Primitive of Arctangent Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21643 | Primitive of Power of x by Arcsine of x | :$\ds \int x^m \arcsin x \rd x = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {1 - x^2} }
| c = Derivative ... | :$\ds \int x^m \arcsin x \rd x = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {1 - x^2}... | Primitive of Power of x by Arcsine of x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x | [
"Primitives involving Inverse Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsine Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21644 | Primitive of Power of x by Arcsine of x | :$\ds \int x^m \arcsin x \rd x = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsin x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = Primitive of $x^m \arcsin x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arcsin \frac x a \rd x
| r =... | :$\ds \int x^m \arcsin x \rd x = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsin x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = [[Primitive of Power of x by Arcsine of x|Primitive of $x^m \arcsin x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l... | Primitive of Power of x by Arcsine of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x_over_a/Proof_1 | [
"Primitives involving Inverse Sine Function"
] | [] | [
"Primitive of Power of x by Arcsine of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21645 | Primitive of Power of x by Arcsine of x | :$\ds \int x^m \arcsin x \rd x = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {a^2 - x^2} }
| c = D... | :$\ds \int x^m \arcsin x \rd x = \frac {x^{m + 1} } {m + 1} \arcsin x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of Power of x by Arcsine of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_x_over_a/Proof_2 | [
"Primitives involving Inverse Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21646 | Characterization of Compact Element in Complete Lattice/Statement 1 implies Statement 3 | Let $L = \struct{S, \preceq}$ be a complete lattice.
Let $a \in S$ be a compact element
Then:
:$\forall A \subseteq S : a \preceq \sup A \implies \exists F \subseteq A : F$ is finite $: a \preceq \sup F$ | Let $A \subseteq S : a \preceq \sup A$.
Let $D = \leftset{b \in A : \exists F \subseteq A : F}$ is finite $\rightset{: b = \sup F}$
==== $D$ is a Directed Subset ====
We show that $D$ is a directed subset.
Let $x, y \in D$.
By definition of $D$:
:$\exists F, G \subseteq A: F, G$ are finite $: x = \sup F, y = \sup G$
Le... | Let $L = \struct{S, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $a \in S$ be a [[Definition:Compact Element|compact element]]
Then:
:$\forall A \subseteq S : a \preceq \sup A \implies \exists F \subseteq A : F$ is [[Definition:Finite Set|finite]] $: a \preceq \sup F$ | Let $A \subseteq S : a \preceq \sup A$.
Let $D = \leftset{b \in A : \exists F \subseteq A : F}$ is [[Definition:Finite Set|finite]] $\rightset{: b = \sup F}$
==== $D$ is a Directed Subset ====
We show that $D$ is a [[Definition:Directed Subset|directed subset]].
Let $x, y \in D$.
By definition of $D$:
:$\exists ... | Characterization of Compact Element in Complete Lattice/Statement 1 implies Statement 3 | https://proofwiki.org/wiki/Characterization_of_Compact_Element_in_Complete_Lattice/Statement_1_implies_Statement_3 | https://proofwiki.org/wiki/Characterization_of_Compact_Element_in_Complete_Lattice/Statement_1_implies_Statement_3 | [
"Characterization of Compact Element in Complete Lattice"
] | [
"Definition:Complete Lattice",
"Definition:Compact Element",
"Definition:Finite Set"
] | [
"Definition:Finite Set",
"Definition:Directed Subset",
"Definition:Finite Set",
"Union of Finite Sets is Finite",
"Definition:Finite Set",
"Definition:Supremum of Set",
"Definition:Directed Subset",
"Definition:Finite Set",
"Supremum of Subset",
"Definition:Upper Bound",
"Definition:Supremum of ... |
proofwiki-21647 | Characterization of Compact Element in Complete Lattice/Statement 3 implies Statement 2 | Let $L = \struct{S, \preceq}$ be a complete lattice.
Let $a \in S$ satisfy:
:$\forall A \subseteq S : a \preceq \sup A \implies \exists F \subseteq A : F$ is finite $: a \preceq \sup F$
Then:
:$\forall I \subseteq S : I$ is an ideal $: a \preceq \sup I \implies a \in I$ | Let:
:$I \subseteq S : I$ is an ideal $: a \preceq \sup I$
We have {{Hypothesis}}:
:$\exists F \subseteq I : F$ is finite $: a \preceq \sup F$
By {{Join-semilattice-ideal-axiom|2}}:
:$\sup F \in I$
By {{Join-semilattice-ideal-axiom|1}}:
:$a \in I$
Since $I$ was arbitrary:
:$\forall I \subseteq S : I$ is an ideal $: a \... | Let $L = \struct{S, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $a \in S$ satisfy:
:$\forall A \subseteq S : a \preceq \sup A \implies \exists F \subseteq A : F$ is [[Definition:Finite Set|finite]] $: a \preceq \sup F$
Then:
:$\forall I \subseteq S : I$ is an [[Definition:Lattice Ideal|idea... | Let:
:$I \subseteq S : I$ is an [[Definition:Lattice Ideal|ideal]] $: a \preceq \sup I$
We have {{Hypothesis}}:
:$\exists F \subseteq I : F$ is [[Definition:Finite Set|finite]] $: a \preceq \sup F$
By {{Join-semilattice-ideal-axiom|2}}:
:$\sup F \in I$
By {{Join-semilattice-ideal-axiom|1}}:
:$a \in I$
Since $I$ wa... | Characterization of Compact Element in Complete Lattice/Statement 3 implies Statement 2 | https://proofwiki.org/wiki/Characterization_of_Compact_Element_in_Complete_Lattice/Statement_3_implies_Statement_2 | https://proofwiki.org/wiki/Characterization_of_Compact_Element_in_Complete_Lattice/Statement_3_implies_Statement_2 | [
"Characterization of Compact Element in Complete Lattice"
] | [
"Definition:Complete Lattice",
"Definition:Finite Set",
"Definition:Lattice Ideal"
] | [
"Definition:Lattice Ideal",
"Definition:Finite Set",
"Definition:Lattice Ideal"
] |
proofwiki-21648 | Characterization of Compact Element in Complete Lattice/Statement 2 implies Statement 1 | Let $L = \struct{S, \preceq}$ be a complete lattice.
Let $a \in S$ satisfy:
:$\forall I \subseteq S : I$ is an ideal $: a \preceq \sup I \implies a \in I$
Then:
:$a$ is a compact element | Let $D$ be a directed subset of $S$:
:$a \preceq \sup D$
Let $I = {b \in S : \exists d \in D : b \preceq d}$.
==== $I$ is an Ideal of $S$ ====
We will show that $I$ satisfies the join semilattice ideal axioms.
===== $I$ is a Lower Section =====
Let $x \in I$.
Let $y \preceq x$.
By definition of $I$:
:$\exists d \in D :... | Let $L = \struct{S, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $a \in S$ satisfy:
:$\forall I \subseteq S : I$ is an [[Definition:Lattice Ideal|ideal]] $: a \preceq \sup I \implies a \in I$
Then:
:$a$ is a [[Definition:Compact Element|compact element]] | Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$:
:$a \preceq \sup D$
Let $I = {b \in S : \exists d \in D : b \preceq d}$.
==== $I$ is an Ideal of $S$ ====
We will show that $I$ satisfies the [[Axiom:Join Semilattice Ideal Axioms|join semilattice ideal axioms]].
===== $I$ is a Lower Section =====
... | Characterization of Compact Element in Complete Lattice/Statement 2 implies Statement 1 | https://proofwiki.org/wiki/Characterization_of_Compact_Element_in_Complete_Lattice/Statement_2_implies_Statement_1 | https://proofwiki.org/wiki/Characterization_of_Compact_Element_in_Complete_Lattice/Statement_2_implies_Statement_1 | [
"Characterization of Compact Element in Complete Lattice"
] | [
"Definition:Complete Lattice",
"Definition:Lattice Ideal",
"Definition:Compact Element"
] | [
"Definition:Directed Subset",
"Axiom:Join Semilattice Ideal Axioms",
"Definition:Lower Section",
"Definition:Directed Subset",
"Definition:Join (Order Theory)",
"Axiom:Join Semilattice Ideal Axioms",
"Definition:Lattice Ideal",
"Definition:Supremum of Set",
"Definition:Upper Bound",
"Definition:Su... |
proofwiki-21649 | Primitive of Power of x by Arccosine of x | :$\ds \int x^m \arccos x \rd x = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqrt {1 - x^2} }
| c = Derivati... | :$\ds \int x^m \arccos x \rd x = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqrt {1 - x... | Primitive of Power of x by Arccosine of x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x | [
"Primitives involving Inverse Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosine Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21650 | Primitive of Power of x by Arccosine of x | :$\ds \int x^m \arccos x \rd x = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccos x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = Primitive of $x^m \arccos x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arccos \frac x a \rd x
| r =... | :$\ds \int x^m \arccos x \rd x = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccos x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = [[Primitive of Power of x by Arccosine of x|Primitive of $x^m \arccos x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn |... | Primitive of Power of x by Arccosine of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x_over_a/Proof_1 | [
"Primitives involving Inverse Cosine Function"
] | [] | [
"Primitive of Power of x by Arccosine of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21651 | Primitive of Power of x by Arccosine of x | :$\ds \int x^m \arccos x \rd x = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqrt {a^2 - x^2} }
| c ... | :$\ds \int x^m \arccos x \rd x = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqr... | Primitive of Power of x by Arccosine of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x_over_a/Proof_2 | [
"Primitives involving Inverse Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21652 | Primitive of Power of x by Arctangent of x | :$\ds \int x^m \arctan x \rd x = \frac {x^{m + 1} } {m + 1} \arctan x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {x^2 + 1}
| c = Derivative of $\arct... | :$\ds \int x^m \arctan x \rd x = \frac {x^{m + 1} } {m + 1} \arctan x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {x^2 + 1}
... | Primitive of Power of x by Arctangent of x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_x | [
"Primitives involving Inverse Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arctangent Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21653 | Primitive of Power of x by Arctangent of x | :$\ds \int x^m \arctan x \rd x = \frac {x^{m + 1} } {m + 1} \arctan x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arctan x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arctan x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}
| c = Primitive of $x^m \arctan x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arctan \frac x a \rd x
| r = \int a^m... | :$\ds \int x^m \arctan x \rd x = \frac {x^{m + 1} } {m + 1} \arctan x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arctan x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arctan x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}
| c = [[Primitive of Power of x by Arctangent of x|Primitive of $x^m \arctan x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \in... | Primitive of Power of x by Arctangent of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_x_over_a/Proof_1 | [
"Primitives involving Inverse Tangent Function"
] | [] | [
"Primitive of Power of x by Arctangent of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21654 | Primitive of Power of x by Arctangent of x | :$\ds \int x^m \arctan x \rd x = \frac {x^{m + 1} } {m + 1} \arctan x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a^2}
| c = Derivative... | :$\ds \int x^m \arctan x \rd x = \frac {x^{m + 1} } {m + 1} \arctan x - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {x^2 + a... | Primitive of Power of x by Arctangent of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_x_over_a/Proof_2 | [
"Primitives involving Inverse Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arctangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21655 | Primitive of Power of x by Arccotangent of x | :$\ds \int x^m \arccot x \rd x = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {x^2 + 1}
| c = Derivative of $\a... | :$\ds \int x^m \arccot x \rd x = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {x^2 + 1}
... | Primitive of Power of x by Arccotangent of x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x | [
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccotangent Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21656 | Primitive of Power of x by Arccotangent of x | :$\ds \int x^m \arccot x \rd x = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccot x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}
| c = Primitive of $x^m \arccot x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arccot \frac x a \rd x
| r = \int a^m... | :$\ds \int x^m \arccot x \rd x = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccot x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}
| c = [[Primitive of Power of x by Arccotangent of x|Primitive of $x^m \arccot x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \... | Primitive of Power of x by Arccotangent of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x_over_a/Proof_1 | [
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Primitive of Power of x by Arccotangent of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21657 | Primitive of Power of x by Arccotangent of x | :$\ds \int x^m \arccot x \rd x = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 + a^2}
| c = Derivat... | :$\ds \int x^m \arccot x \rd x = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 ... | Primitive of Power of x by Arccotangent of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x_over_a/Proof_2 | [
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccotangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21658 | Primitive of Power of x by Arcsecant of x | :$\ds \int x^m \arcsec x \rd x = \begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x... | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \dfrac 1 {x \sqrt {x^2 - 1^2} } & :... | :$\ds \int x^m \arcsec x \rd x = \begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x... | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \dfrac 1 ... | Primitive of Power of x by Arcsecant of x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x | [
"Primitives involving Inverse Secant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsecant Function/Corollary 1",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21659 | Primitive of Power of x by Arcsecant of x | :$\ds \int x^m \arcsec x \rd x = \begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x... | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsec x \rd x
| r = <nowiki>\begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfrac... | :$\ds \int x^m \arcsec x \rd x = \begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x... | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsec x \rd x
| r = <nowiki>\begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfra... | Primitive of Power of x by Arcsecant of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x_over_a/Proof_1 | [
"Primitives involving Inverse Secant Function"
] | [] | [
"Primitive of Power of x by Arcsecant of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21660 | Primitive of Power of x by Arcsecant of x | :$\ds \int x^m \arcsec x \rd x = \begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x... | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin {cases} \dfrac a {x \sqrt ... | :$\ds \int x^m \arcsec x \rd x = \begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x... | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin ... | Primitive of Power of x by Arcsecant of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x_over_a/Proof_2 | [
"Primitives involving Inverse Secant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arcsecant Function/Corollary 1",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21661 | Primitive of Power of x by Arccosecant of x | :$\ds \int x^m \arccsc x \rd x = \begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc ... | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \dfrac {-1} {x \sqrt {x^2 - 1} } &... | :$\ds \int x^m \arccsc x \rd x = \begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc ... | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \dfrac {... | Primitive of Power of x by Arccosecant of x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x | [
"Primitives involving Inverse Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosecant Function",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21662 | Primitive of Power of x by Arccosecant of x | :$\ds \int x^m \arccsc x \rd x = \begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc ... | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccsc x \rd x
| r = <nowiki>\begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\ \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \df... | :$\ds \int x^m \arccsc x \rd x = \begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc ... | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccsc x \rd x
| r = <nowiki>\begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\ \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \d... | Primitive of Power of x by Arccosecant of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x_over_a/Proof_1 | [
"Primitives involving Inverse Cosecant Function"
] | [] | [
"Primitive of Power of x by Arccosecant of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21663 | Primitive of Power of x by Arccosecant of x | :$\ds \int x^m \arccsc x \rd x = \begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc ... | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin {cases} \dfrac {-a} {x \s... | :$\ds \int x^m \arccsc x \rd x = \begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc ... | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin... | Primitive of Power of x by Arccosecant of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x_over_a/Proof_2 | [
"Primitives involving Inverse Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arccosecant Function/Corollary",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-21664 | Primitive of Power of x by Arcsine of a x | :$\ds \int x^n \arcsin a x \rd x = \frac {x^{n + 1} } {n + 1} \arcsin a x - \frac a {n + 1} \int \frac {x^{n + 1} \rd x} {\sqrt {1 - a^2 x^2} }$
for $n \ne -1$. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^n \arcsin x \rd x
| r = \frac {x^{n + 1} } {n + 1} \arcsin x - \frac 1 {n + 1} \int \frac {x^{n + 1} \rd x} {\sqrt {1 - x^2} }
| c = Primitive of $x^n \arcsin x$
}}
{{end-eqn}}
Let:
{{begin-eqn}}
{{eqn | l = u
| r = a x
}}
{{eqn | ll = \leadsto
... | :$\ds \int x^n \arcsin a x \rd x = \frac {x^{n + 1} } {n + 1} \arcsin a x - \frac a {n + 1} \int \frac {x^{n + 1} \rd x} {\sqrt {1 - a^2 x^2} }$
for $n \ne -1$. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^n \arcsin x \rd x
| r = \frac {x^{n + 1} } {n + 1} \arcsin x - \frac 1 {n + 1} \int \frac {x^{n + 1} \rd x} {\sqrt {1 - x^2} }
| c = [[Primitive of Power of x by Arcsine of x|Primitive of $x^n \arcsin x$]]
}}
{{end-eqn}}
Let:
{{begin-eqn}}
{{eqn | l... | Primitive of Power of x by Arcsine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsine_of_a_x | [
"Primitives involving Inverse Sine Function"
] | [] | [
"Primitive of Power of x by Arcsine of x",
"Integration by Substitution"
] |
proofwiki-21665 | Primitive of Power of x by Arccosine of a x | :$\ds \int x^n \arccos a x \rd x = \frac {x^{n + 1} } {n + 1} \arccos a x + \frac a {n + 1} \int \frac {x^{n + 1} \rd x} {\sqrt {1 - a^2 x^2} }$
for $n \ne -1$. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^n \arccos x \rd x
| r = \frac {x^{n + 1} } {n + 1} \arccos x + \frac 1 {n + 1} \int \frac {x^{n + 1} \rd x} {\sqrt {1 - x^2} }
| c = Primitive of $x^n \arccos x$
}}
{{end-eqn}}
Let:
{{begin-eqn}}
{{eqn | l = u
| r = a x
}}
{{eqn | ll = \leadsto
... | :$\ds \int x^n \arccos a x \rd x = \frac {x^{n + 1} } {n + 1} \arccos a x + \frac a {n + 1} \int \frac {x^{n + 1} \rd x} {\sqrt {1 - a^2 x^2} }$
for $n \ne -1$. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^n \arccos x \rd x
| r = \frac {x^{n + 1} } {n + 1} \arccos x + \frac 1 {n + 1} \int \frac {x^{n + 1} \rd x} {\sqrt {1 - x^2} }
| c = [[Primitive of Power of x by Arccosine of x|Primitive of $x^n \arccos x$]]
}}
{{end-eqn}}
Let:
{{begin-eqn}}
{{eqn |... | Primitive of Power of x by Arccosine of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_a_x | [
"Primitives involving Inverse Cosine Function"
] | [] | [
"Primitive of Power of x by Arccosine of x",
"Integration by Substitution"
] |
proofwiki-21666 | Primitive of Power of x by Arctangent of a x | :$\ds \int x^n \arctan a x \rd x = \frac {x^{n + 1} } {n + 1} \arctan a x + \frac a {n + 1} \int \frac {x^{n + 1} \rd x} {a^2 x^2 + 1}$
for $n \ne -1$. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^n \arctan x \rd x
| r = \frac {x^{n + 1} } {n + 1} \arctan x - \frac 1 {n + 1} \int \frac {x^{n + 1} \rd x} {x^2 + 1}
| c = Primitive of $x^n \arctan x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^n \arctan a x \rd x
| r = \int \dfrac 1 ... | :$\ds \int x^n \arctan a x \rd x = \frac {x^{n + 1} } {n + 1} \arctan a x + \frac a {n + 1} \int \frac {x^{n + 1} \rd x} {a^2 x^2 + 1}$
for $n \ne -1$. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^n \arctan x \rd x
| r = \frac {x^{n + 1} } {n + 1} \arctan x - \frac 1 {n + 1} \int \frac {x^{n + 1} \rd x} {x^2 + 1}
| c = [[Primitive of Power of x by Arctangent of x|Primitive of $x^n \arctan x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \in... | Primitive of Power of x by Arctangent of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arctangent_of_a_x | [
"Primitives involving Inverse Tangent Function"
] | [] | [
"Primitive of Power of x by Arctangent of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21667 | Primitive of General Exponential of a x | :$\ds \int b^{a x} \rd x = \frac {b^{a x} } {a \ln b} + C$
where:
:$a \ne 0$
:$b > 0, b \ne 1$ | {{begin-eqn}}
{{eqn | l = \int b^x \rd x
| r = \frac {b^x} {\ln b} + C
| c = Primitive of $b^x$
}}
{{eqn | ll= \leadsto
| l = \int b^{a x} \rd x
| r = \frac 1 a \paren {\frac {b^x} {\ln b} } + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {b^{a x} } {a \ln b} + C
... | :$\ds \int b^{a x} \rd x = \frac {b^{a x} } {a \ln b} + C$
where:
:$a \ne 0$
:$b > 0, b \ne 1$ | {{begin-eqn}}
{{eqn | l = \int b^x \rd x
| r = \frac {b^x} {\ln b} + C
| c = [[Primitive of General Exponential Function|Primitive of $b^x$]]
}}
{{eqn | ll= \leadsto
| l = \int b^{a x} \rd x
| r = \frac 1 a \paren {\frac {b^x} {\ln b} } + C
| c = [[Primitive of Function of Constant Multipl... | Primitive of General Exponential of a x | https://proofwiki.org/wiki/Primitive_of_General_Exponential_of_a_x | https://proofwiki.org/wiki/Primitive_of_General_Exponential_of_a_x | [
"Primitive of General Exponential of a x",
"Primitives involving Exponential Function"
] | [] | [
"Primitive of Exponential Function/General Result",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21668 | Reduction Formula for Power of x by Exponential of a x | :$\ds \int x^n e^{a x} \rd x = \frac {x^n e^{a x} } a - \dfrac n a \int x^{n - 1} e^{a x} \rd x$
for $n \in \Z_{>0}$, $a \in \R_{\ne 0}$. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^n
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n x^{n - 1}
| c = Power Rule for Derivatives
}}
{{... | :$\ds \int x^n e^{a x} \rd x = \frac {x^n e^{a x} } a - \dfrac n a \int x^{n - 1} e^{a x} \rd x$
for $n \in \Z_{>0}$, $a \in \R_{\ne 0}$. | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^n
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n x^{n - 1}
|... | Reduction Formula for Power of x by Exponential of a x | https://proofwiki.org/wiki/Reduction_Formula_for_Power_of_x_by_Exponential_of_a_x | https://proofwiki.org/wiki/Reduction_Formula_for_Power_of_x_by_Exponential_of_a_x | [
"Primitive of Power of x by Exponential of a x",
"Primitives involving Exponential Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Power Rule for Derivatives",
"Primitive of Exponential of a x",
"Integration by Parts"
] |
proofwiki-21669 | Reduction Formula for Power of x by General Exponential of a x | :$\ds \int x^n b^{a x} \rd x = \frac {x^n b^{a x} } {a \ln b} - \dfrac n {a \ln b} \int x^{n - 1} b^{a x} \rd x$
for $n \in \Z_{>0}$, $a \in \R_{\ne 0}$, $b > 0$, $b \ne 1$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^n
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n x^{n - 1}
| c = Power Rule for Derivatives
}}
{{... | :$\ds \int x^n b^{a x} \rd x = \frac {x^n b^{a x} } {a \ln b} - \dfrac n {a \ln b} \int x^{n - 1} b^{a x} \rd x$
for $n \in \Z_{>0}$, $a \in \R_{\ne 0}$, $b > 0$, $b \ne 1$ | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^n
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n x^{n - 1}
|... | Reduction Formula for Power of x by General Exponential of a x | https://proofwiki.org/wiki/Reduction_Formula_for_Power_of_x_by_General_Exponential_of_a_x | https://proofwiki.org/wiki/Reduction_Formula_for_Power_of_x_by_General_Exponential_of_a_x | [
"Primitives involving Exponential Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Power Rule for Derivatives",
"Primitive of General Exponential of a x",
"Integration by Parts"
] |
proofwiki-21670 | Primitive of Logarithm of a x | :$\ds \int \ln a x \rd x = x \ln a x - x + C$ | {{begin-eqn}}
{{eqn | l = \int \ln x \rd x
| r = x \ln x - x + C
| c = Primitive of $\ln x$
}}
{{eqn | ll= \leadsto
| l = \int \ln a x \rd x
| r = \frac 1 a \paren {a x \ln a x - a x} + C
| c = Primitive of Function of Constant Multiple
}}
{{eqn | r = x \ln a x - x + C
| c = simplify... | :$\ds \int \ln a x \rd x = x \ln a x - x + C$ | {{begin-eqn}}
{{eqn | l = \int \ln x \rd x
| r = x \ln x - x + C
| c = [[Primitive of Logarithm of x|Primitive of $\ln x$]]
}}
{{eqn | ll= \leadsto
| l = \int \ln a x \rd x
| r = \frac 1 a \paren {a x \ln a x - a x} + C
| c = [[Primitive of Function of Constant Multiple]]
}}
{{eqn | r = x ... | Primitive of Logarithm of a x | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_a_x | https://proofwiki.org/wiki/Primitive_of_Logarithm_of_a_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Primitive of Logarithm of x",
"Primitive of Function of Constant Multiple"
] |
proofwiki-21671 | Derivative of Natural Logarithm of a x | Let $\ln x$ be the natural logarithm function.
Then:
:$\map {\dfrac \d {\d x} } {\ln a x} = \dfrac 1 x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\ln a x}
| r = a \map {\dfrac \d {\d \paren {a x} } } {\ln a x}
| c = Derivative of Function of Constant Multiple
}}
{{eqn | r = a \dfrac 1 {a x}
| c = Derivative of Natural Logarithm
}}
{{eqn | r = \dfrac 1 x
| c =
}}
{{end-eqn}}
Category:Der... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm function]].
Then:
:$\map {\dfrac \d {\d x} } {\ln a x} = \dfrac 1 x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\ln a x}
| r = a \map {\dfrac \d {\d \paren {a x} } } {\ln a x}
| c = [[Derivative of Function of Constant Multiple]]
}}
{{eqn | r = a \dfrac 1 {a x}
| c = [[Derivative of Natural Logarithm]]
}}
{{eqn | r = \dfrac 1 x
| c =
}}
{{end-eqn}}
[[C... | Derivative of Natural Logarithm of a x | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_of_a_x | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_of_a_x | [
"Derivatives involving Logarithm Functions"
] | [
"Definition:Natural Logarithm"
] | [
"Derivative of Function of Constant Multiple",
"Derivative of Natural Logarithm Function",
"Category:Derivatives involving Logarithm Functions"
] |
proofwiki-21672 | Primitive of Power of x by Logarithm of a x | :$\ds \int x^n \ln a x \rd x = \dfrac {x^{n + 1} } {n + 1} \ln a x - \dfrac {x^{n + 1} } {\paren {n + 2}^2} + C$
for $n \ne -1$. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \ln a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = Derivative of $\ln a x$
}}
{{e... | :$\ds \int x^n \ln a x \rd x = \dfrac {x^{n + 1} } {n + 1} \ln a x - \dfrac {x^{n + 1} } {\paren {n + 2}^2} + C$
for $n \ne -1$. | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \ln a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 x
| c = [[De... | Primitive of Power of x by Logarithm of a x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Logarithm_of_a_x | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Logarithm_of_a_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Definition:Primitive",
"Derivative of Natural Logarithm of a x",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-21673 | Primitive of Reciprocal of x by Logarithm of a x | :$\ds \int \frac {\d x} {x \ln a x} = \ln \size {\ln a x} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \ln a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = \frac 1 x
| c = Derivative of $\ln a x$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \ln a x}
| r = \ln \size {\ln a x} + C
| c = Primitive of Function under its... | :$\ds \int \frac {\d x} {x \ln a x} = \ln \size {\ln a x} + C$ | {{begin-eqn}}
{{eqn | l = z
| r = \ln a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = \frac 1 x
| c = [[Derivative of Natural Logarithm of a x|Derivative of $\ln a x$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \ln a x}
| r = \ln \size {\ln a x} + C
... | Primitive of Reciprocal of x by Logarithm of a x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Logarithm_of_a_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Logarithm_of_a_x | [
"Primitives involving Logarithm Function"
] | [] | [
"Derivative of Natural Logarithm of a x",
"Primitive of Function under its Derivative"
] |
proofwiki-21674 | Primitive of Exponential of a x by Hyperbolic Sine of b x/Exponential Form | :$\ds \int e^{a x} \sinh b x \rd x = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} - \frac {e^{-b x} } {a - b} } + C$
for $a^2 \ne b^2$. | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sinh b x \rd x
| r = \int e^{a x} \paren {\frac {e^{b x} - e^{-b x} } 2} \rd x
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \paren {e^{b x} - e^{-b x} } \rd x
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 2 \int... | :$\ds \int e^{a x} \sinh b x \rd x = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} - \frac {e^{-b x} } {a - b} } + C$
for $a^2 \ne b^2$. | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sinh b x \rd x
| r = \int e^{a x} \paren {\frac {e^{b x} - e^{-b x} } 2} \rd x
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \paren {e^{b x} - e^{-b x} } \rd x
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = \frac 1 2 ... | Primitive of Exponential of a x by Hyperbolic Sine of b x/Exponential Form | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Hyperbolic_Sine_of_b_x/Exponential_Form | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Hyperbolic_Sine_of_b_x/Exponential_Form | [
"Primitive of Exponential of a x by Hyperbolic Sine of b x"
] | [] | [
"Primitive of Constant Multiple of Function",
"Exponent Combination Laws/Product of Powers",
"Linear Combination of Integrals/Indefinite",
"Primitive of Exponential of a x",
"Exponent Combination Laws/Product of Powers"
] |
proofwiki-21675 | Primitive of Exponential of a x by Hyperbolic Sine of b x/Hyperbolic Form | :$\ds \int e^{a x} \sinh b x \rd x = \frac {e^{a x} \paren {a \sinh b x - b \cosh b x} } {a^2 - b^2} + C$
for $a^2 \ne b^2$. | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sinh b x \rd x
| r = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} - \frac {e^{-b x} } {a - b} } + C
| c = Primitive of $e^{a x} \sinh b x$: Exponential Form
}}
{{eqn | r = \frac {e^{a x} } 2 \paren {\frac {e^{b x} \paren {a - b} } {\paren {a + b} \paren {a - b}... | :$\ds \int e^{a x} \sinh b x \rd x = \frac {e^{a x} \paren {a \sinh b x - b \cosh b x} } {a^2 - b^2} + C$
for $a^2 \ne b^2$. | {{begin-eqn}}
{{eqn | l = \int e^{a x} \sinh b x \rd x
| r = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} - \frac {e^{-b x} } {a - b} } + C
| c = [[Primitive of Exponential of a x by Hyperbolic Sine of b x/Exponential Form|Primitive of $e^{a x} \sinh b x$: Exponential Form]]
}}
{{eqn | r = \frac {e^{... | Primitive of Exponential of a x by Hyperbolic Sine of b x/Hyperbolic Form | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Hyperbolic_Sine_of_b_x/Hyperbolic_Form | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Hyperbolic_Sine_of_b_x/Hyperbolic_Form | [
"Primitive of Exponential of a x by Hyperbolic Sine of b x"
] | [] | [
"Primitive of Exponential of a x by Hyperbolic Sine of b x/Exponential Form",
"Difference of Two Squares"
] |
proofwiki-21676 | Primitive of Exponential of a x by Hyperbolic Cosine of b x/Exponential Form | :$\ds \int e^{a x} \cosh b x \rd x = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} + \frac {e^{-b x} } {a - b} } + C$
for $a^2 \ne b^2$. | {{begin-eqn}}
{{eqn | l = \int e^{a x} \cosh b x \rd x
| r = \int e^{a x} \paren {\frac {e^{b x} + e^{-b x} } 2} \rd x
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \paren {e^{b x} + e^{-b x} } \rd x
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 2 \i... | :$\ds \int e^{a x} \cosh b x \rd x = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} + \frac {e^{-b x} } {a - b} } + C$
for $a^2 \ne b^2$. | {{begin-eqn}}
{{eqn | l = \int e^{a x} \cosh b x \rd x
| r = \int e^{a x} \paren {\frac {e^{b x} + e^{-b x} } 2} \rd x
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \paren {e^{b x} + e^{-b x} } \rd x
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = \frac 1 ... | Primitive of Exponential of a x by Hyperbolic Cosine of b x/Exponential Form | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Hyperbolic_Cosine_of_b_x/Exponential_Form | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Hyperbolic_Cosine_of_b_x/Exponential_Form | [
"Primitive of Exponential of a x by Hyperbolic Cosine of b x"
] | [] | [
"Primitive of Constant Multiple of Function",
"Exponent Combination Laws/Product of Powers",
"Linear Combination of Integrals/Indefinite",
"Primitive of Exponential of a x",
"Exponent Combination Laws/Product of Powers"
] |
proofwiki-21677 | Primitive of Exponential of a x by Hyperbolic Cosine of b x/Hyperbolic Form | :$\ds \int e^{a x} \cosh b x \rd x = \frac {e^{a x} \paren {a \cosh b x + b \sinh b x} } {a^2 - b^2} + C$
for $a^2 \ne b^2$. | {{begin-eqn}}
{{eqn | l = \int e^{a x} \cosh b x \rd x
| r = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} + \frac {e^{-b x} } {a - b} } + C
| c = Primitive of $e^{a x} \cosh b x$: Exponential Form
}}
{{eqn | r = \frac {e^{a x} } 2 \paren {\frac {e^{b x} \paren {a - b} } {\paren {a + b} \paren {a - b}... | :$\ds \int e^{a x} \cosh b x \rd x = \frac {e^{a x} \paren {a \cosh b x + b \sinh b x} } {a^2 - b^2} + C$
for $a^2 \ne b^2$. | {{begin-eqn}}
{{eqn | l = \int e^{a x} \cosh b x \rd x
| r = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} + \frac {e^{-b x} } {a - b} } + C
| c = [[Primitive of Exponential of a x by Hyperbolic Cosine of b x/Exponential Form|Primitive of $e^{a x} \cosh b x$: Exponential Form]]
}}
{{eqn | r = \frac {e... | Primitive of Exponential of a x by Hyperbolic Cosine of b x/Hyperbolic Form | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Hyperbolic_Cosine_of_b_x/Hyperbolic_Form | https://proofwiki.org/wiki/Primitive_of_Exponential_of_a_x_by_Hyperbolic_Cosine_of_b_x/Hyperbolic_Form | [
"Primitive of Exponential of a x by Hyperbolic Cosine of b x"
] | [] | [
"Primitive of Exponential of a x by Hyperbolic Cosine of b x/Exponential Form",
"Difference of Two Squares"
] |
proofwiki-21678 | Definite Integral of Exponential of minus a x squared from 0 to Infinity | :$\ds \int_0^\infty \map \exp {-a x^2} \rd x = \dfrac 1 2 \sqrt {\dfrac \pi a}$
for $a > 0$. | Recall Integral to Infinity of $\map \exp {-x^2}$:
:$\ds \int_0^\infty \map \exp {-x^2} \rd x = \dfrac {\sqrt \pi} 2$
Then:
{{begin-eqn}}
{{eqn | l = \int \map \exp {-a x^2} \rd x
| r = \int \map \exp {-\paren {\sqrt a x}^2} \rd x
| c =
}}
{{eqn | r = \dfrac 1 {\sqrt a} \int \map \exp {-\paren {\sqrt a x}^... | :$\ds \int_0^\infty \map \exp {-a x^2} \rd x = \dfrac 1 2 \sqrt {\dfrac \pi a}$
for $a > 0$. | Recall [[Integral to Infinity of Exponential of -t^2|Integral to Infinity of $\map \exp {-x^2}$]]:
:$\ds \int_0^\infty \map \exp {-x^2} \rd x = \dfrac {\sqrt \pi} 2$
Then:
{{begin-eqn}}
{{eqn | l = \int \map \exp {-a x^2} \rd x
| r = \int \map \exp {-\paren {\sqrt a x}^2} \rd x
| c =
}}
{{eqn | r = \dfr... | Definite Integral of Exponential of minus a x squared from 0 to Infinity | https://proofwiki.org/wiki/Definite_Integral_of_Exponential_of_minus_a_x_squared_from_0_to_Infinity | https://proofwiki.org/wiki/Definite_Integral_of_Exponential_of_minus_a_x_squared_from_0_to_Infinity | [
"Integral to Infinity of Exponential of -t^2",
"Gauss Error Function",
"Definite Integrals involving Exponential Function"
] | [] | [
"Integral to Infinity of Exponential of -t^2",
"Primitive of Function of Constant Multiple",
"Definition:Definite Integral/Limits of Integration"
] |
proofwiki-21679 | Euler Characteristic is not Dependent upon Subdivision | Let $S$ be a surface.
Let $T_1$ and $T_2$ be subdivisions of $S$.
Let:
:$\map {\chi_1} S$ be the Euler characteristic of $S$ as calculated using $T_1$
:$\map {\chi_2} S$ be the Euler characteristic of $S$ as calculated using $T_2$.
Then:
:$\map {\chi_1} S = \map {\chi_2} S$ | Recall:
{{begin-eqn}}
{{eqn | l = \map {\chi_1} S
| r = \map {\chi_2} S
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map v {T_1} - \map e {T_1} + \map f {T_1}
| r = \map v {T_2} - \map e {T_2} + \map f {T_2}
| c = {{Defof|Euler Characteristic of Surface}}
}}
{{end-eqn}}
where:
:$\map v {T_... | Let $S$ be a [[Definition:Surface|surface]].
Let $T_1$ and $T_2$ be [[Definition:Subdivision of Surface|subdivisions]] of $S$.
Let:
:$\map {\chi_1} S$ be the [[Definition:Euler Characteristic of Surface|Euler characteristic]] of $S$ as calculated using $T_1$
:$\map {\chi_2} S$ be the [[Definition:Euler Characteristic... | Recall:
{{begin-eqn}}
{{eqn | l = \map {\chi_1} S
| r = \map {\chi_2} S
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map v {T_1} - \map e {T_1} + \map f {T_1}
| r = \map v {T_2} - \map e {T_2} + \map f {T_2}
| c = {{Defof|Euler Characteristic of Surface}}
}}
{{end-eqn}}
where:
:$\map v {... | Euler Characteristic is not Dependent upon Subdivision | https://proofwiki.org/wiki/Euler_Characteristic_is_not_Dependent_upon_Subdivision | https://proofwiki.org/wiki/Euler_Characteristic_is_not_Dependent_upon_Subdivision | [
"Euler Characteristic",
"Subdivisions of Surfaces"
] | [
"Definition:Surface",
"Definition:Subdivision of Surface",
"Definition:Euler Characteristic of Surface",
"Definition:Euler Characteristic of Surface"
] | [
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Size",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Planar Graph/Face"
] |
proofwiki-21680 | Relative Pseudocomplement Preserves Order | Let $\struct {S, \vee, \wedge, \preceq}$ be a Brouwerian lattice.
Let $a, b, c \in S$.
Let $b \preceq c$.
Then
:$a \to b \preceq a \to c$
where $x \to y$ denotes the relative pseudocomplement of $x$ with respect to $y$. | We have:
{{begin-eqn}}
{{eqn | l = a \to b
| r = \max \set {a \wedge x : x \in S : a \wedge x \preceq b}
| c = {{Defof|Relative Pseudocomplement}}
}}
{{eqn | o = \preceq
| r = \max \set {a \wedge x : x \in S : a \wedge x \preceq c}
| c = Finer Supremum Precedes Supremum
}}
{{eqn | r = a \to c
... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Brouwerian Lattice|Brouwerian lattice]].
Let $a, b, c \in S$.
Let $b \preceq c$.
Then
:$a \to b \preceq a \to c$
where $x \to y$ denotes the [[Definition:Relative Pseudocomplement|relative pseudocomplement]] of $x$ with respect to $y$. | We have:
{{begin-eqn}}
{{eqn | l = a \to b
| r = \max \set {a \wedge x : x \in S : a \wedge x \preceq b}
| c = {{Defof|Relative Pseudocomplement}}
}}
{{eqn | o = \preceq
| r = \max \set {a \wedge x : x \in S : a \wedge x \preceq c}
| c = [[Finer Supremum Precedes Supremum]]
}}
{{eqn | r = a \to ... | Relative Pseudocomplement Preserves Order | https://proofwiki.org/wiki/Relative_Pseudocomplement_Preserves_Order | https://proofwiki.org/wiki/Relative_Pseudocomplement_Preserves_Order | [
"Brouwerian Lattices"
] | [
"Definition:Brouwerian Lattice",
"Definition:Relative Pseudocomplement"
] | [
"Finer Supremum Precedes Supremum",
"Category:Brouwerian Lattices"
] |
proofwiki-21681 | Eulerian Graph is Semi-Eulerian | Let $G$ be an Eulerian graph.
Then $G$ is also a semi-Eulerian graph. | Recall the definition of Eulerian graph:
{{:Definition:Eulerian Graph}}
Recall the definition of semi-Eulerian graph:
{{:Definition:Semi-Eulerian Graph}}
So, let $G$ be an Eulerian graph.
Let $C$ be an Eulerian circuit for $G$.
From Eulerian Circuit is Eulerian Trail, $C$ is also an Eulerian trail for $G$.
Thus $G$ ha... | Let $G$ be an [[Definition:Eulerian Graph|Eulerian graph]].
Then $G$ is also a [[Definition:Semi-Eulerian Graph|semi-Eulerian graph]]. | Recall the definition of [[Definition:Eulerian Graph|Eulerian graph]]:
{{:Definition:Eulerian Graph}}
Recall the definition of [[Definition:Semi-Eulerian Graph|semi-Eulerian graph]]:
{{:Definition:Semi-Eulerian Graph}}
So, let $G$ be an [[Definition:Eulerian Graph|Eulerian graph]].
Let $C$ be an [[Definition:Euleria... | Eulerian Graph is Semi-Eulerian | https://proofwiki.org/wiki/Eulerian_Graph_is_Semi-Eulerian | https://proofwiki.org/wiki/Eulerian_Graph_is_Semi-Eulerian | [
"Eulerian Graphs",
"Semi-Eulerian Graphs"
] | [
"Definition:Eulerian Graph",
"Definition:Semi-Eulerian Graph"
] | [
"Definition:Eulerian Graph",
"Definition:Semi-Eulerian Graph",
"Definition:Eulerian Graph",
"Definition:Eulerian Circuit",
"Eulerian Circuit is Eulerian Trail",
"Definition:Eulerian Trail",
"Definition:Eulerian Trail",
"Definition:Semi-Eulerian Graph",
"Category:Eulerian Graphs",
"Category:Semi-Eu... |
proofwiki-21682 | Eulerian Circuit is Eulerian Trail | Let $G$ be a graph.
Let $C$ be an Eulerian circuit for $G$.
Then $C$ is also an Eulerian trail for $G$. | Recall the definition of Eulerian circuit:
{{:Definition:Eulerian Circuit}}
Recall the definition of Eulerian trail:
{{:Definition:Eulerian Trail}}
Recall the definition of circuit:
{{:Definition:Circuit (Graph Theory)}}
Hence an Eulerian circuit is an instance of an Eulerian trail.
{{qed}}
Category:Eulerian Circuits
C... | Let $G$ be a [[Definition:Graph (Graph Theory)|graph]].
Let $C$ be an [[Definition:Eulerian Circuit|Eulerian circuit]] for $G$.
Then $C$ is also an [[Definition:Eulerian Trail|Eulerian trail]] for $G$. | Recall the definition of [[Definition:Eulerian Circuit|Eulerian circuit]]:
{{:Definition:Eulerian Circuit}}
Recall the definition of [[Definition:Eulerian Trail|Eulerian trail]]:
{{:Definition:Eulerian Trail}}
Recall the definition of [[Definition:Circuit (Graph Theory)|circuit]]:
{{:Definition:Circuit (Graph Theory)... | Eulerian Circuit is Eulerian Trail | https://proofwiki.org/wiki/Eulerian_Circuit_is_Eulerian_Trail | https://proofwiki.org/wiki/Eulerian_Circuit_is_Eulerian_Trail | [
"Eulerian Circuits",
"Eulerian Trails"
] | [
"Definition:Graph (Graph Theory)",
"Definition:Eulerian Circuit",
"Definition:Eulerian Trail"
] | [
"Definition:Eulerian Circuit",
"Definition:Eulerian Trail",
"Definition:Circuit (Graph Theory)",
"Definition:Eulerian Circuit",
"Definition:Eulerian Trail",
"Category:Eulerian Circuits",
"Category:Eulerian Trails"
] |
proofwiki-21683 | Euler's Criterion | Let $a$ be a residue order $n$ of $m$, where $a$ and $m$ are coprime.
Then:
:$a^{\map \phi m / d} \equiv 1 \pmod m$
where:
:$\map \phi m$ denotes the Euler $\phi$ function of $m$
:$d$ denotes the gretest common divisor of $\map \phi m$ and $n$
:$\equiv$ denotes modulo congruence. | {{ProofWanted}}
{{Namedfor|Leonhard Paul Euler|cat = Euler}} | Let $a$ be a [[Definition:Residue (Number Theory)|residue order $n$]] of $m$, where $a$ and $m$ are [[Definition:Coprime Integers|coprime]].
Then:
:$a^{\map \phi m / d} \equiv 1 \pmod m$
where:
:$\map \phi m$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]] of $m$
:$d$ denotes the [[Definition:Great... | {{ProofWanted}}
{{Namedfor|Leonhard Paul Euler|cat = Euler}} | Euler's Criterion | https://proofwiki.org/wiki/Euler's_Criterion | https://proofwiki.org/wiki/Euler's_Criterion | [
"Euler's Criterion",
"Residues (Number Theory)",
"Number Theory"
] | [
"Definition:Residue (Number Theory)",
"Definition:Coprime/Integers",
"Definition:Euler Phi Function",
"Definition:Greatest Common Divisor",
"Definition:Congruence (Number Theory)/Integers"
] | [] |
proofwiki-21684 | Euler's Criterion | Let $a$ be a residue order $n$ of $m$, where $a$ and $m$ are coprime.
Then:
:$a^{\map \phi m / d} \equiv 1 \pmod m$
where:
:$\map \phi m$ denotes the Euler $\phi$ function of $m$
:$d$ denotes the gretest common divisor of $\map \phi m$ and $n$
:$\equiv$ denotes modulo congruence. | Trivially, any $a \not \equiv 0 \pmod p$ is either a quadratic residue or a quadratic non-residue, modulo $p$.
Therefore, it suffices to check the sufficient condition for both of the equations (i.e., the ''if'' parts from the ''iff''s).
So let $a$ be a quadratic non-residue of $p$.
Also, let $b \in \set {1, 2, \ldots,... | Let $a$ be a [[Definition:Residue (Number Theory)|residue order $n$]] of $m$, where $a$ and $m$ are [[Definition:Coprime Integers|coprime]].
Then:
:$a^{\map \phi m / d} \equiv 1 \pmod m$
where:
:$\map \phi m$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]] of $m$
:$d$ denotes the [[Definition:Great... | Trivially, any $a \not \equiv 0 \pmod p$ is either a [[Definition:Quadratic Residue|quadratic residue]] or a [[Definition:Quadratic Non-Residue|quadratic non-residue]], modulo $p$.
Therefore, it suffices to check the sufficient condition for both of the equations (i.e., the ''if'' parts from the ''iff''s).
So let $a... | Euler's Criterion/Quadratic Residue/Proof 1 | https://proofwiki.org/wiki/Euler's_Criterion | https://proofwiki.org/wiki/Euler's_Criterion/Quadratic_Residue/Proof_1 | [
"Euler's Criterion",
"Residues (Number Theory)",
"Number Theory"
] | [
"Definition:Residue (Number Theory)",
"Definition:Coprime/Integers",
"Definition:Euler Phi Function",
"Definition:Greatest Common Divisor",
"Definition:Congruence (Number Theory)/Integers"
] | [
"Definition:Quadratic Residue",
"Definition:Quadratic Residue/Non-Residue",
"Definition:Quadratic Residue/Non-Residue",
"Definition:Congruence (Number Theory)",
"Solution of Linear Congruence",
"Definition:Quadratic Residue",
"Definition:Residue Class",
"Wilson's Theorem",
"Definition:Quadratic Resi... |
proofwiki-21685 | Euler's Criterion | Let $a$ be a residue order $n$ of $m$, where $a$ and $m$ are coprime.
Then:
:$a^{\map \phi m / d} \equiv 1 \pmod m$
where:
:$\map \phi m$ denotes the Euler $\phi$ function of $m$
:$d$ denotes the gretest common divisor of $\map \phi m$ and $n$
:$\equiv$ denotes modulo congruence. | First note that the square roots of $1$ are $1, -1 \pmod p$.
Also, we have that $a^{p - 1} \equiv 1 \pmod p$ by Fermat's Little Theorem.
Combining these two observations, we find:
:$a^{\frac {p - 1} 2} \equiv 1 \text{ or } -1 \pmod p$
The theorem is therefore equivalent to stating that $a$ is a quadratic residue modulo... | Let $a$ be a [[Definition:Residue (Number Theory)|residue order $n$]] of $m$, where $a$ and $m$ are [[Definition:Coprime Integers|coprime]].
Then:
:$a^{\map \phi m / d} \equiv 1 \pmod m$
where:
:$\map \phi m$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]] of $m$
:$d$ denotes the [[Definition:Great... | First note that the [[Square Root of 1 Mod Prime|square roots of $1$]] are $1, -1 \pmod p$.
Also, we have that $a^{p - 1} \equiv 1 \pmod p$ by [[Fermat's Little Theorem]].
Combining these two observations, we find:
:$a^{\frac {p - 1} 2} \equiv 1 \text{ or } -1 \pmod p$
The theorem is therefore equivalent to statin... | Euler's Criterion/Quadratic Residue/Proof 2 | https://proofwiki.org/wiki/Euler's_Criterion | https://proofwiki.org/wiki/Euler's_Criterion/Quadratic_Residue/Proof_2 | [
"Euler's Criterion",
"Residues (Number Theory)",
"Number Theory"
] | [
"Definition:Residue (Number Theory)",
"Definition:Coprime/Integers",
"Definition:Euler Phi Function",
"Definition:Greatest Common Divisor",
"Definition:Congruence (Number Theory)/Integers"
] | [
"Square Root of 1 Mod Prime",
"Fermat's Little Theorem",
"Definition:Quadratic Residue",
"Definition:Quadratic Residue/Non-Residue",
"Definition:Congruence (Number Theory)",
"Definition:Quadratic Residue",
"Congruence of Powers",
"Fermat's Little Theorem",
"Definition:Primitive Root (Number Theory)"... |
proofwiki-21686 | Element Well Inside Itself Iff Has Complement | :$\forall a \in S : a \eqslantless a \iff a$ has a complement | Follows immediately from:
:* Definition:Well Inside Relation
:* Definition:Complement (Lattice Theory)
{{qed}} | :$\forall a \in S : a \eqslantless a \iff a$ has a [[Definition:Complement (Lattice Theory)|complement]] | Follows immediately from:
:* [[Definition:Well Inside Relation]]
:* [[Definition:Complement (Lattice Theory)]]
{{qed}} | Element Well Inside Itself Iff Has Complement | https://proofwiki.org/wiki/Element_Well_Inside_Itself_Iff_Has_Complement | https://proofwiki.org/wiki/Element_Well_Inside_Itself_Iff_Has_Complement | [
"Well Inside Relation"
] | [
"Definition:Complement (Lattice Theory)"
] | [
"Definition:Well Inside Relation",
"Definition:Complement (Lattice Theory)"
] |
proofwiki-21687 | Equivalence of Definitions of Even Permutation | {{TFAE|def = Even Permutation}}
Let $n \in \N$ be a natural number.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\rho \in S_n$ be a permutation in $S_n$. | The '''sign of $\rho$''' is defined as:
:$\map \sgn \rho = \begin {cases} 1 & : \text {$k$ even} \\ -1 & : \text {$k$ odd} \\ \end {cases}$
The result follows.
{{qed}}
Category:Even Permutations
foox8y9d0yzpalw87h0e8vo4fvcpt1s | {{TFAE|def = Even Permutation}}
Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $\rho \in S_n$ be a [[Definition:Permutation on n Letters|permutation in $S_n$]]. | The '''[[Definition:Sign of Permutation on n Letters|sign of $\rho$]]''' is defined as:
:$\map \sgn \rho = \begin {cases} 1 & : \text {$k$ even} \\ -1 & : \text {$k$ odd} \\ \end {cases}$
The result follows.
{{qed}}
[[Category:Even Permutations]]
foox8y9d0yzpalw87h0e8vo4fvcpt1s | Equivalence of Definitions of Even Permutation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Even_Permutation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Even_Permutation | [
"Even Permutations"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group/n Letters",
"Definition:Permutation on n Letters"
] | [
"Definition:Sign of Permutation on n Letters",
"Category:Even Permutations"
] |
proofwiki-21688 | Well Inside Implies Predecessor | :$\forall a,b \in S : a \eqslantless b \implies a \preceq b$ | Let $a, b \in S : a \eqslantless b$.
By definition of well inside relation:
:$\exists c \in S : a \wedge c = \bot, b \vee c = \top$
We have:
{{begin-eqn}}
{{eqn | l = a
| r = a \wedge \top
| c = Predecessor is Infimum
}}
{{eqn | r = a \wedge \paren{b \vee c}
| c = By choice of $c$
}}
{{eqn | r = \par... | :$\forall a,b \in S : a \eqslantless b \implies a \preceq b$ | Let $a, b \in S : a \eqslantless b$.
By definition of [[Definition:Well Inside Relation|well inside relation]]:
:$\exists c \in S : a \wedge c = \bot, b \vee c = \top$
We have:
{{begin-eqn}}
{{eqn | l = a
| r = a \wedge \top
| c = [[Predecessor is Infimum]]
}}
{{eqn | r = a \wedge \paren{b \vee c}
... | Well Inside Implies Predecessor | https://proofwiki.org/wiki/Well_Inside_Implies_Predecessor | https://proofwiki.org/wiki/Well_Inside_Implies_Predecessor | [
"Well Inside Relation"
] | [] | [
"Definition:Well Inside Relation",
"Predecessor is Infimum",
"Successor is Supremum",
"Predecessor is Infimum"
] |
proofwiki-21689 | Well Inside Relation Extends to Predecessor and Successor | :$\forall a,b,c,d \in S : a \preceq b \eqslantless c \preceq d \implies a \eqslantless d$ | Let $a,b,c,d \in S : a \preceq b \eqslantless c \preceq d$
By definition of well inside relation:
:$\exists x \in S : b \wedge x = \bot, c \vee x = \top$
We have:
{{begin-eqn}}
{{eqn | l = \bot
| o = \preceq
| r = a \wedge x
| c = {{Defof|Smallest Element}}
}}
{{eqn | o = \preceq
| r = b \wedge... | :$\forall a,b,c,d \in S : a \preceq b \eqslantless c \preceq d \implies a \eqslantless d$ | Let $a,b,c,d \in S : a \preceq b \eqslantless c \preceq d$
By definition of [[Definition:Well Inside Relation|well inside relation]]:
:$\exists x \in S : b \wedge x = \bot, c \vee x = \top$
We have:
{{begin-eqn}}
{{eqn | l = \bot
| o = \preceq
| r = a \wedge x
| c = {{Defof|Smallest Element}}
}}
... | Well Inside Relation Extends to Predecessor and Successor | https://proofwiki.org/wiki/Well_Inside_Relation_Extends_to_Predecessor_and_Successor | https://proofwiki.org/wiki/Well_Inside_Relation_Extends_to_Predecessor_and_Successor | [
"Well Inside Relation"
] | [] | [
"Definition:Well Inside Relation",
"Infimum Precedes Coarser Infimum",
"Finer Supremum Precedes Supremum",
"Definition:Well Inside Relation"
] |
proofwiki-21690 | Well Inside Elements Form Filter | :$\forall a \in S : \set{b \in S: a \eqslantless b}$ is a lattice filter | Let $a \in S$.
Let $F = \set{b \in S: a \eqslantless b}$. | :$\forall a \in S : \set{b \in S: a \eqslantless b}$ is a [[Definition:Lattice Filter|lattice filter]] | Let $a \in S$.
Let $F = \set{b \in S: a \eqslantless b}$. | Well Inside Elements Form Filter | https://proofwiki.org/wiki/Well_Inside_Elements_Form_Filter | https://proofwiki.org/wiki/Well_Inside_Elements_Form_Filter | [
"Well Inside Relation"
] | [
"Definition:Lattice Filter"
] | [] |
proofwiki-21691 | Elements Well Inside Form Ideal | :$\forall a \in S : \set{b \in S: b \eqslantless a}$ is a a lattice ideal | Let $a \in S$.
Let $I = \set{b \in S: b \eqslantless a}$. | :$\forall a \in S : \set{b \in S: b \eqslantless a}$ is a a [[Definition:Lattice Ideal|lattice ideal]] | Let $a \in S$.
Let $I = \set{b \in S: b \eqslantless a}$. | Elements Well Inside Form Ideal | https://proofwiki.org/wiki/Elements_Well_Inside_Form_Ideal | https://proofwiki.org/wiki/Elements_Well_Inside_Form_Ideal | [
"Well Inside Relation"
] | [
"Definition:Lattice Ideal"
] | [] |
proofwiki-21692 | Even Derivatives of Cotangent of Pi Z at One Fourth | :$ \ds \valueat {\dfrac {\d^{2 n} } {\d z^{2 n} } \cot \pi z} {z \mathop = \frac 1 4} = \paren {-1}^n \paren {2 \pi}^{2 n} E_{2 n}$
where:
:$E_n$ denotes the $n$th Euler number
:$n$ is a non-negative integer. | === Lemma ===
{{:Even Derivatives of Cotangent of Pi Z at One Fourth/Lemma}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \map \tan {\dfrac \pi 4 + z}
| r = \map \sec {2 \pi z} + \map \tan {2 \pi z}
| c = Lemma
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {E_{2 n} \paren {2 \pi z}^{2 n} } {\pare... | :$ \ds \valueat {\dfrac {\d^{2 n} } {\d z^{2 n} } \cot \pi z} {z \mathop = \frac 1 4} = \paren {-1}^n \paren {2 \pi}^{2 n} E_{2 n}$
where:
:$E_n$ denotes the $n$th [[Definition:Euler Numbers|Euler number]]
:$n$ is a [[Definition:Non-Negative Integer|non-negative integer]]. | === [[Even Derivatives of Cotangent of Pi Z at One Fourth/Lemma|Lemma]] ===
{{:Even Derivatives of Cotangent of Pi Z at One Fourth/Lemma}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \map \tan {\dfrac \pi 4 + z}
| r = \map \sec {2 \pi z} + \map \tan {2 \pi z}
| c = [[Even Derivatives of Cotangent of Pi Z at On... | Even Derivatives of Cotangent of Pi Z at One Fourth | https://proofwiki.org/wiki/Even_Derivatives_of_Cotangent_of_Pi_Z_at_One_Fourth | https://proofwiki.org/wiki/Even_Derivatives_of_Cotangent_of_Pi_Z_at_One_Fourth | [
"Euler Numbers"
] | [
"Definition:Euler Numbers",
"Definition:Positive/Integer"
] | [
"Even Derivatives of Cotangent of Pi Z at One Fourth/Lemma",
"Even Derivatives of Cotangent of Pi Z at One Fourth/Lemma",
"Power Series Expansion for Secant Function",
"Power Series Expansion for Tangent Function"
] |
proofwiki-21693 | Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 5 | :$\map \beta 5 = \dfrac {5 \pi^5} {1536} $ | {{begin-eqn}}
{{eqn | l = \map \beta {2 n + 1}
| r = \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}
| c = Dirichlet Beta Function at Odd Positive Integers
}}
{{eqn | ll = \leadsto
| l = \map \beta 5
| r = \paren {-1}^2 \dfrac {E_4 \pi^5 } {4^3 \paren {4}!}
| c = se... | :$\map \beta 5 = \dfrac {5 \pi^5} {1536} $ | {{begin-eqn}}
{{eqn | l = \map \beta {2 n + 1}
| r = \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}
| c = [[Dirichlet Beta Function at Odd Positive Integers]]
}}
{{eqn | ll = \leadsto
| l = \map \beta 5
| r = \paren {-1}^2 \dfrac {E_4 \pi^5 } {4^3 \paren {4}!}
| c ... | Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 5 | https://proofwiki.org/wiki/Dirichlet_Beta_Function_at_Odd_Positive_Integers/Examples/Dirichlet_Beta_Function_of_5 | https://proofwiki.org/wiki/Dirichlet_Beta_Function_at_Odd_Positive_Integers/Examples/Dirichlet_Beta_Function_of_5 | [
"Dirichlet Beta Function at Odd Positive Integers",
"Examples of Dirichlet Beta Function Values"
] | [] | [
"Dirichlet Beta Function at Odd Positive Integers",
"Definition:Euler Numbers",
"Category:Dirichlet Beta Function at Odd Positive Integers",
"Category:Examples of Dirichlet Beta Function Values"
] |
proofwiki-21694 | Whole Sample Space represents Certain Event | Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
The sample space $\Sigma$ represents an event which is certain. | By definition, an event is a subset of the sample space $\Omega$.
Hence an outcome of $\EE$ is necessarily an element of $\Omega$.
That is, the probability that $\omega \in \Omega$ is $1$.
The result follows by definition of certain event. | Let the [[Definition:Probability Space|probability space]] of an [[Definition:Experiment|experiment]] $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
The [[Definition:Sample Space|sample space]] $\Sigma$ represents an [[Definition:Event|event]] which is [[Definition:Certain Event|certain]]. | By definition, an [[Definition:Event|event]] is a [[Definition:Subset|subset]] of the [[Definition:Sample Space|sample space]] $\Omega$.
Hence an [[Definition:Outcome|outcome]] of $\EE$ is necessarily an [[Definition:Element|element]] of $\Omega$.
That is, the [[Definition:Probability|probability]] that $\omega \in \... | Whole Sample Space represents Certain Event | https://proofwiki.org/wiki/Whole_Sample_Space_represents_Certain_Event | https://proofwiki.org/wiki/Whole_Sample_Space_represents_Certain_Event | [
"Certain Events",
"Sample Spaces"
] | [
"Definition:Probability Space",
"Definition:Experiment",
"Definition:Sample Space",
"Definition:Event",
"Definition:Event/Occurrence/Certainty"
] | [
"Definition:Event",
"Definition:Subset",
"Definition:Sample Space",
"Definition:Elementary Event",
"Definition:Element",
"Definition:Probability",
"Definition:Event/Occurrence/Certainty"
] |
proofwiki-21695 | Universal Statement has no Existential Import | A universal statement of the form:
:''All $A$ are $B$
has no '''existential import'''. | If there exist no $A$, then:
:''All $A$ are $B$
is vacuously true, and hence remains true.
If there exist no $B$, then:
:''All $A$ are $B$
is vacuously true when there exist no $A$.
Hence the result by definition of '''existential import'''.
{{qed}} | A [[Definition:Universal Statement|universal statement]] of the form:
:''All $A$ are $B$
has no '''[[Definition:Existential Import|existential import]]'''. | If there exist no $A$, then:
:''All $A$ are $B$
is [[Definition:Vacuous Truth|vacuously true]], and hence remains [[Definition:True|true]].
If there exist no $B$, then:
:''All $A$ are $B$
is [[Definition:Vacuous Truth|vacuously true]] when there exist no $A$.
Hence the result by definition of '''[[Definition:Existe... | Universal Statement has no Existential Import | https://proofwiki.org/wiki/Universal_Statement_has_no_Existential_Import | https://proofwiki.org/wiki/Universal_Statement_has_no_Existential_Import | [
"Universal Statement has no Existential Import",
"Universal Quantifier",
"Existential Import"
] | [
"Definition:Universal Statement",
"Definition:Existential Import"
] | [
"Definition:Vacuous Truth",
"Definition:True",
"Definition:Vacuous Truth",
"Definition:Existential Import"
] |
proofwiki-21696 | Polylogarithm of Square | :$\map {\Li_s} z + \map {\Li_s} {-z} = 2^{1 - s} \map {\Li_s} {z^2}$ | <onlyinclude>
{{begin-eqn}}
{{eqn | l = \map {\Li_s} z + \map {\Li_s} {-z}
| r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^s} + \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} {n^s}
| c = {{Defof|Polylogarithm}}
}}
{{eqn | r = \paren {z + \frac {z^2} {2^s} + \frac {z^3} {3^s} + \frac {z^4} {4^s} + \frac ... | :$\map {\Li_s} z + \map {\Li_s} {-z} = 2^{1 - s} \map {\Li_s} {z^2}$ | <onlyinclude>
{{begin-eqn}}
{{eqn | l = \map {\Li_s} z + \map {\Li_s} {-z}
| r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^s} + \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} {n^s}
| c = {{Defof|Polylogarithm}}
}}
{{eqn | r = \paren {z + \frac {z^2} {2^s} + \frac {z^3} {3^s} + \frac {z^4} {4^s} + \frac ... | Polylogarithm of Square | https://proofwiki.org/wiki/Polylogarithm_of_Square | https://proofwiki.org/wiki/Polylogarithm_of_Square | [
"Polylogarithm"
] | [] | [
"Definition:Odd Integer",
"Definition:Even Integer",
"Category:Polylogarithm"
] |
proofwiki-21697 | Recurrence Relation for Polylogarithms | :$\ds \map {\Li_{s + 1} } z = \int_0^z \dfrac {\map {\Li_s} t} t \rd t$ | {{begin-eqn}}
{{eqn | l = \int_0^z \dfrac {\map {\Li_s} t} t \rd t
| r = \int_0^z \frac 1 t \times \sum_{n \mathop = 1}^\infty \frac {t^n} {n^s} \rd t
| c = {{Defof|Polylogarithm}}
}}
{{eqn | r = \int_0^z \sum_{n \mathop = 1}^\infty \frac {t^{n - 1} } {n^s} \rd t
| c = Quotient of Powers
}}
{{eqn | r = \... | :$\ds \map {\Li_{s + 1} } z = \int_0^z \dfrac {\map {\Li_s} t} t \rd t$ | {{begin-eqn}}
{{eqn | l = \int_0^z \dfrac {\map {\Li_s} t} t \rd t
| r = \int_0^z \frac 1 t \times \sum_{n \mathop = 1}^\infty \frac {t^n} {n^s} \rd t
| c = {{Defof|Polylogarithm}}
}}
{{eqn | r = \int_0^z \sum_{n \mathop = 1}^\infty \frac {t^{n - 1} } {n^s} \rd t
| c = [[Quotient of Powers]]
}}
{{eqn | r... | Recurrence Relation for Polylogarithms | https://proofwiki.org/wiki/Recurrence_Relation_for_Polylogarithms | https://proofwiki.org/wiki/Recurrence_Relation_for_Polylogarithms | [
"Polylogarithm",
"Recurrence Relations"
] | [] | [
"Exponent Combination Laws/Quotient of Powers",
"Fubini's Theorem",
"Primitive of Power",
"Category:Polylogarithm",
"Category:Recurrence Relations"
] |
proofwiki-21698 | Exponential Distribution is Special Case of Gamma Distribution | The exponential distribution is a special case of the gamma distribution. | {{ProofWanted|Need to rationalise the definitions}} | The [[Definition:Exponential Distribution|exponential distribution]] is a special case of the [[Definition:Gamma Distribution|gamma distribution]]. | {{ProofWanted|Need to rationalise the definitions}} | Exponential Distribution is Special Case of Gamma Distribution | https://proofwiki.org/wiki/Exponential_Distribution_is_Special_Case_of_Gamma_Distribution | https://proofwiki.org/wiki/Exponential_Distribution_is_Special_Case_of_Gamma_Distribution | [
"Exponential Distribution",
"Gamma Distribution"
] | [
"Definition:Exponential Distribution",
"Definition:Gamma Distribution"
] | [] |
proofwiki-21699 | Rate of Exponential Growth | Let $y = a e^{b t}$ be an exponential growth function.
Then the rate of growth of $y$ is proportional to the value of $y$ such that:
:$\dfrac {\d y} {\d t} = b y$ | {{begin-eqn}}
{{eqn | l = y
| r = a e^{b t}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d y} {\d t}
| r = a b e^{b t}
| c = {{Corollary|Derivative of Exponential Function|1}}
}}
{{eqn | r = b y
| c = Definition of $y$
}}
{{end-eqn}}
{{qed}} | Let $y = a e^{b t}$ be an [[Definition:Exponential Growth|exponential growth function]].
Then the [[Definition:Rate of Change|rate]] of growth of $y$ is [[Definition:Proportion|proportional]] to the [[Definition:Value of Variable|value]] of $y$ such that:
:$\dfrac {\d y} {\d t} = b y$ | {{begin-eqn}}
{{eqn | l = y
| r = a e^{b t}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d y} {\d t}
| r = a b e^{b t}
| c = {{Corollary|Derivative of Exponential Function|1}}
}}
{{eqn | r = b y
| c = Definition of $y$
}}
{{end-eqn}}
{{qed}} | Rate of Exponential Growth | https://proofwiki.org/wiki/Rate_of_Exponential_Growth | https://proofwiki.org/wiki/Rate_of_Exponential_Growth | [
"Exponential Growth"
] | [
"Definition:Exponential Growth",
"Definition:Rate of Change",
"Definition:Proportion",
"Definition:Variable/Value"
] | [] |
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