rag/knowledge_base/integration_basics.md

Integration Basics

Standard Integrals

• $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$
• $\int \frac{1}{x} , dx = \ln|x| + C$
• $\int e^x , dx = e^x + C$
• $\int a^x , dx = \frac{a^x}{\ln a} + C$
• $\int \sin x , dx = -\cos x + C$
• $\int \cos x , dx = \sin x + C$
• $\int \sec^2 x , dx = \tan x + C$
• $\int \csc^2 x , dx = -\cot x + C$
• $\int \frac{1}{\sqrt{1-x^2}} , dx = \sin^{-1} x + C$
• $\int \frac{1}{1+x^2} , dx = \tan^{-1} x + C$

Integration by Substitution

$\int f(g(x)) g'(x) , dx = \int f(u) , du$ where $u = g(x)$

Integration by Parts

$\int u , dv = uv - \int v , du$

LIATE priority for choosing $u$: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential

Partial Fractions

$\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$ $\frac{P(x)}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}$

Definite Integrals Properties

• $\int_a^b f(x) , dx = -\int_b^a f(x) , dx$
• $\int_a^b f(x) , dx = \int_a^c f(x) , dx + \int_c^b f(x) , dx$
• $\int_0^a f(x) , dx = \int_0^a f(a-x) , dx$
• If $f$ is even: $\int_{-a}^a f(x) , dx = 2\int_0^a f(x) , dx$
• If $f$ is odd: $\int_{-a}^a f(x) , dx = 0$