id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-7c51e05020d327be0a3bc82cbdb955eadacce286e3046f8196b81a8a9a5baa7e | Calculate the integral $$\int \frac{1}{x^3+12} \, dx$$ without decomposing the fraction into the sum of simpler fractions. | [
" \\frac{1}{12^{2/3}} \\left( \\frac{1}{3} \\ln\\left|x + 12^{1/3}\\right| - \\frac{1}{6} \\ln\\left|x^2 - 12^{1/3}x + 12^{2/3}\\right| + \\frac{1}{\\sqrt{3}} \\arctan\\left( \\frac{2x - 12^{1/3}}{12^{1/3}\\sqrt{3}} \\right) \\right) + C "
] | [
"Calculate the integral $$\\int \\frac{1}{x^3+12} \\, dx$$ without decomposing the fraction into the sum of simpler fractions."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-d1eaef39fa6978ff8b05116e0259428893f92848d4c6142709a1412be0f52c04 | Calculate the integral $$\int \frac{1}{x^6\sqrt{x^2-1}}dx.$$ | [
"\\dfrac{(8x^4 + 4x^2 + 3)\\sqrt{x^2 - 1}}{15x^5} + C"
] | [
"Calculate the integral $$\\int \\frac{1}{x^6\\sqrt{x^2-1}}dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e5c557adb38d65a7378d83e5b5e642540639ed8b2a5b361497b33f595da8986 | Calculate the integral $$\int \frac{\ln \sqrt{x^2+1}}{x^2} \, dx.$$ | [
"\\(-\\frac{\\ln(x^2+1)}{2x} + \\arctan{x} + C\\)"
] | [
"Calculate the integral\n$$\\int \\frac{\\ln \\sqrt{x^2+1}}{x^2} \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-57d32a1199288c259fb9e7e4f0ca1d895ca2daf92de51f6393b1f146503658f6 | Calculate the integral $$\int \frac{\sin^2x}{\sin^3x+\cos^3x}dx$$ | [
"\\frac{1}{2\\sqrt{3}} \\ln\\left| \\frac{\\sqrt{3} + \\sin x + \\cos x}{\\sqrt{3} - \\sin x - \\cos x} \\right| + \\frac{1}{2} \\ln|\\sin x + \\cos x| + C"
] | [
"Calculate the integral\n$$\\int \\frac{\\sin^2x}{\\sin^3x+\\cos^3x}dx$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-02eec2bfcd67f0abb8195eae6d1381e1d85c2b36c466ebd7dead80096b644355 | Calculate the integral $$\int \frac{\sqrt{1+x^4}}{x^4-1} \, dx$$ | [
"\\text{The integral does not have an elementary antiderivative.}"
] | [
"Calculate the integral\n$$\\int \\frac{\\sqrt{1+x^4}}{x^4-1} \\, dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-183b28f10bec0c08fdf1f08c981f0701ec6eee58f81b9779d5c1bf792fc985d1 | Calculate the integral $$\int \frac{dx}{\sin (x - 1) \sin (x - 2)}.$$ Hint: Change $\tan x = t$. | [
"\\(\\frac{1}{\\sin 1} \\ln \\left| \\frac{\\sin(x-2)}{\\sin(x-1)} \\right| + C\\)"
] | [
"Calculate the integral $$\\int \\frac{dx}{\\sin (x - 1) \\sin (x - 2)}.$$ Hint: Change $\\tan x = t$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-bfea81d055c8e3d75173d550eea99b124db561fd52ed96626744e1150e751277 | Calculate the integral $$\int \frac{e^{\sin x}}{x} \, dx$$ | [
"\\int \\frac{e^{\\sin x}}{x} \\, dx"
] | [
"Calculate the integral\n$$\\int \\frac{e^{\\sin x}}{x} \\, dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-b6618e34383205d6e21af231ddb9330bbdc887b559634c6d7aa578deb492b5a4 | Calculate the integral $$\int \frac{y^4+16}{y^6+64} dy$$ | [
"\\(\\frac{1}{6} \\left[ \\tan^{-1} \\left( \\frac{2y}{4-y^2} \\right) - 2 \\tan^{-1} \\left( \\frac{2}{y} \\right) \\right] + C\\)"
] | [
"Calculate the integral\n\n$$\\int \\frac{y^4+16}{y^6+64} dy$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1172d8d28a2f17b2b208a62fb60c667af651c6116315ec6ce3a663f9bec42d65 | Calculate the integral $$\int \ln|\sin{x}|\, dx.$$ | [
" -x \\ln 2 - \\frac{1}{2} \\operatorname{Cl}_2(2x) + C "
] | [
"Calculate the integral $$\\int \\ln|\\sin{x}|\\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f2e7ff7a7d8a6e97c6e3fc863c4b081f491842a98c25d4be13003b286e3d31eb | Calculate the integral $$\int \sqrt[5]{\tan x}\, dx$$ in terms of elementary functions. | [
"\\[\n\\frac{1}{8} \\left(2 \\sqrt{10-2 \\sqrt{5}} \\tan ^{-1}\\left(\\frac{\\sqrt{10-2 \\sqrt{5}}-4 \\sqrt[5]{\\tan (x)}}{1+\\sqrt{5}}\\right)+2 \\sqrt{2 \\left(5+\\sqrt{5}\\right)} \\tan\n ^{-1}\\left(\\frac{\\sqrt{2 \\left(5+\\sqrt{5}\\right)}-4 \\sqrt[5]{\\tan (x)}}{1-\\sqrt{5}}\\right)\\right.\\\\\n \\left... | [
"Calculate the integral $$\\int \\sqrt[5]{\\tan x}\\, dx$$ in terms of elementary functions."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-6396e7d28c439417a76226de8ade7dbb3ef3b4f06e64d3cb3c7a8a7892486848 | Calculate the integral $$\int \sqrt{1-\ln^2x} \, dx$$ | [
"\\frac{x}{2} \\left( \\sqrt{1 - (\\ln x)^2} + \\arcsin(\\ln x) \\right) + C"
] | [
"Calculate the integral\n$$\\int \\sqrt{1-\\ln^2x} \\, dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-7fbb9fcd68a333aa1cb9fc1a5e7f5c2c634008b77836cd8763453f43909b3011 | Calculate the integral $$\int \sqrt{\frac{1+\cos x}{1-\cos x} \cdot \frac{1+\cos \psi \cos x}{1-\cos \psi \cos x} }\, dx.$$ | [
"\\(2\\left[\\tan^{-1} {\\sqrt{\\frac{1+\\cos \\psi \\cos x}{1-\\cos \\psi \\cos x}}} + \\cot \\frac{\\psi}{2}\\tan^{-1}\\left({\\sqrt{\\frac{1+\\cos \\psi \\cos x}{1-\\cos \\psi \\cos x}}}\n \\cdot \\tan \\frac{\\psi}{2}\\right)\\right] + C\\)"
] | [
"Calculate the integral\n\n$$\\int \\sqrt{\\frac{1+\\cos x}{1-\\cos x} \\cdot \\frac{1+\\cos \\psi \\cos x}{1-\\cos \\psi \\cos x} }\\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9191d6297b01274d6e821e55c58a486ab24622d2e7fbf9f6d0b2501c603bc1af | Calculate the integral $$\int e^{-x}\sec^{2}(x)dx.$$ | [
"\\(-e^{-x} \\sec(x) + C\\)"
] | [
"Calculate the integral $$\\int e^{-x}\\sec^{2}(x)dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ceae04c0daa64588d0bdcbe0c1171a9e486b513af7060bc50c4db8fd8693370 | Calculate the integral $$\int x^a \cdot \ln x \, dx$$ where \( a \in \mathbb{R} \). | [
"For \\( a \\neq -1 \\): \\(\\frac{x^{a+1}}{(a+1)^2} \\left( (a+1) \\ln x - 1 \\right) + C\\)\nFor \\( a = -1 \\): \\(\\frac{1}{2} (\\ln x)^2 + C\\)"
] | [
"Calculate the integral $$\\int x^a \\cdot \\ln x \\, dx$$ where \\( a \\in \\mathbb{R} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-90e27d385cb23cee0d4337edf551a74501e7adc6696554dbb160b7b74d2bfbad | Calculate the integral $$\int\frac{2x^{12}+5x^{9}}{(x^5+x^3+1)^3}dx.$$ | [
"\\(\\frac{x^{10}}{2(x^5 + x^3 + 1)^2} + C\\)"
] | [
"Calculate the integral $$\\int\\frac{2x^{12}+5x^{9}}{(x^5+x^3+1)^3}dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 3
} | 17 | 0 | 0 |
omr-ws-47a8f194a69c130c2bf7000755e7badb6c2cda81b17de37d99aa2240a138c27a | Calculate the integral $$\int\frac{\ln^2x}{x-1}dx$$ | [
"\\ln^2 x \\ln(1 - x) + 2 \\ln x \\, \\text{Li}_2(x) - 2 \\, \\text{Li}_3(x) + C"
] | [
"Calculate the integral\n$$\\int\\frac{\\ln^2x}{x-1}dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-577455fe7f7087b905d0fa48be2c356b1758ccef9a204515c32c312c509e579a | Calculate the integral $$\int\limits_0^{\infty}{\frac{e^{-ax^2}-e^{-bx^2}}{x}\sin{x^2} \, dx}$$ | [
"\\(\\frac{\\tan ^{-1}b-\\tan ^{-1}a}{2}\\)"
] | [
"Calculate the integral\n$$\\int\\limits_0^{\\infty}{\\frac{e^{-ax^2}-e^{-bx^2}}{x}\\sin{x^2} \\, dx}$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-b4bc0b5786b86ff983da9161a1f0f0ae8d07320d35ca4dfb632b8e524e3c9e01 | Calculate the integral $$\int\limits_{-4}^{-1}\frac{x}{x-2}\frac{dx}{\sqrt[3]{(x+4)(x-1)^2}}$$ | [
"\\(0.7306\\)"
] | [
"Calculate the integral\n$$\\int\\limits_{-4}^{-1}\\frac{x}{x-2}\\frac{dx}{\\sqrt[3]{(x+4)(x-1)^2}}$$"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-2893420b82c617b664789e9eb97f44b223af6e3e6a34a724aa1873b7af8c40eb | Calculate the integral $$\int\limits_{-\infty }^{\infty }\frac{(2x-3)\cos x}{x^2-4x+5}dx$$ using Laplace transforms. | [
"\\pi e^{-1} (\\cos 2 - 2 \\sin 2)"
] | [
"Calculate the integral\n$$\\int\\limits_{-\\infty }^{\\infty }\\frac{(2x-3)\\cos x}{x^2-4x+5}dx$$\nusing Laplace transforms."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-4788c4d229b35792c3bc115fd6448cbf647bd0e97d1978a6459e7692f71edadc | Calculate the integral $$\int\limits_{-\infty }^{\infty}\frac{\left | \sin x-\cos x \right |}{x^2+1}dx$$ | [
""
] | [
"Calculate the integral\n$$\\int\\limits_{-\\infty }^{\\infty}\\frac{\\left | \\sin x-\\cos x \\right |}{x^2+1}dx$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-c9ff7b165aa1f3d4c03c63377459f759025e8149af6c5c6a1e37b332003295a0 | Calculate the integral $$\int\limits_{-\pi/2}^{\pi/2}e^{a\sin x}\cos^2 x \, dx$$ | [
"\\(\\frac{\\pi}{a}I_1(a)\\) or \\(\\frac{\\pi}{2}\\sum_{k=0}^\\infty\\frac{a^{2n}}{2^{2n}n!(n+1)!}\\)"
] | [
"Calculate the integral\n$$\\int\\limits_{-\\pi/2}^{\\pi/2}e^{a\\sin x}\\cos^2 x \\, dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 12
} | 34 | 0 | 0 |
omr-ws-407a8d34ff336a2b17ccf87fc751686dbad93c76f793321fb3332cb865256c4a | Calculate the integral $$\int\limits_{0}^{1}dx\int\limits_{x}^{1}dy\int\limits_{y}^{1}e^{z^3}dz$$ | [
"\\(\\frac{e-1}{6}\\)"
] | [
"Calculate the integral\n$$\\int\\limits_{0}^{1}dx\\int\\limits_{x}^{1}dy\\int\\limits_{y}^{1}e^{z^3}dz$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-e41ba9fd25ea64a50cbc33bdcecd3cd6c5c5277755f7b9d9bddc4ce7ec31ada0 | Calculate the integral $$\int\limits_{0}^{1}{\ln \left( -\ln x \right) \operatorname{Li}^{2}(x) \frac{dx}{x}}.$$ | [
"\\zeta'(3) - \\gamma \\zeta(3)"
] | [
"Calculate the integral $$\\int\\limits_{0}^{1}{\\ln \\left( -\\ln x \\right) \\operatorname{Li}^{2}(x) \\frac{dx}{x}}.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-315d6e90a74c5a7f73a0add2ae009496b6b3ce05e2521f234e5f4edc76a22c19 | Calculate the integral $$\int\limits_{0}^{2\pi}\int\limits_{0}^{1}\int\limits_{0}^{1}xy\sqrt{x^2+y^2-2xy\cos \theta } \; dxdyd\theta $$ | [
"\\dfrac{4\\pi}{15}"
] | [
"Calculate the integral\n$$\\int\\limits_{0}^{2\\pi}\\int\\limits_{0}^{1}\\int\\limits_{0}^{1}xy\\sqrt{x^2+y^2-2xy\\cos \\theta } \\; dxdyd\\theta $$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5973dc3df51dbbbe75b3f0513524af5e46084593c73ab9d6d69acfd332c604f1 | Calculate the integral $$\int\limits_{0}^{3}\frac{\sqrt{\arcsin \frac{x}{3}}}{\sqrt[4]{9-x^2}}dx$$ | [
"\\frac{3 \\sqrt{\\pi} \\, \\Gamma\\left(\\frac{3}{4}\\right)^2}{2 \\sqrt{2}}"
] | [
"Calculate the integral\n$$\\int\\limits_{0}^{3}\\frac{\\sqrt{\\arcsin \\frac{x}{3}}}{\\sqrt[4]{9-x^2}}dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-944dcd6d380f1ef44cecdb132b17af2ed63cdc5d8dd3be9a53e493bb62ecbad4 | Calculate the integral $$\int\limits_{0}^{\infty }\frac{\sin \pi x}{x}\arccos\left ( \frac{1}{\sqrt{1+\cos^2x}} \right )dx$$ | [
"\\dfrac{\\pi^2}{8}"
] | [
"Calculate the integral\n$$\\int\\limits_{0}^{\\infty }\\frac{\\sin \\pi x}{x}\\arccos\\left ( \\frac{1}{\\sqrt{1+\\cos^2x}} \\right )dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-15bd175ce10a92429ee32229d14f501c99128350d379de040a24d7563f03dcce | Calculate the integral $$\int\limits_{0}^{\infty }\frac{x\sin^4 x}{\cos x + \cosh x} \, dx$$ | [
"\\(\\frac{176}{225}\\)"
] | [
"Calculate the integral\n$$\\int\\limits_{0}^{\\infty }\\frac{x\\sin^4 x}{\\cos x + \\cosh x} \\, dx$$"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-9cf596c869340d5039e4fdf5b93f0004bd3befb7a2f7e3e5a1c5f29c84132ab1 | Calculate the integral $$\int\limits_{0}^{\infty }\frac{x\sin^4 x}{\cos x+\cosh x}dx$$ | [
"\\dfrac{\\pi^2}{16}"
] | [
"Calculate the integral\n$$\\int\\limits_{0}^{\\infty }\\frac{x\\sin^4 x}{\\cos x+\\cosh x}dx$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-e48656e2190ede86f886b42484765b0b748315232becacaac85b6d813993bea6 | Calculate the integral $$\int\limits_{0}^{\pi/2}\sin \sqrt{\tan x} \, dx$$ | [
"\\(\\pi \\, e^{-\\frac{1}{\\sqrt{2}}} \\sin \\frac{1}{\\sqrt{2}}\\)"
] | [
"Calculate the integral\n$$\\int\\limits_{0}^{\\pi/2}\\sin \\sqrt{\\tan x} \\, dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-960c4e93b02b052d99f752764615444859dbdee87b98eb6d78785c539b6e8583 | Calculate the integral $$\int\limits_{0}^{\sqrt{\pi}}dy\int\limits_{y}^{\sqrt{\pi}}\sin x^2 \, dx$$ | [
"1"
] | [
"Calculate the integral\n$$\\int\\limits_{0}^{\\sqrt{\\pi}}dy\\int\\limits_{y}^{\\sqrt{\\pi}}\\sin x^2 \\, dx$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-14db0eed3978c9124ce9d4496cf104b540601d92f311931712066cc5e242255d | Calculate the integral $$\int\limits_{1}^{+\infty}\frac{\arctan(x-1)}{\sqrt [3] {(x-1)^4}}dx.$$ | [
"\\(\\pi\\sqrt{3}\\)"
] | [
"Calculate the integral\n$$\\int\\limits_{1}^{+\\infty}\\frac{\\arctan(x-1)}{\\sqrt [3] {(x-1)^4}}dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 28
} | 50 | 0 | 0 |
omr-ws-ab73950a081c56396c71aba078eec9666a73cb21f33eaa89a33d043c0384d4b3 | Calculate the integral $$\int^{1}_{0} \frac{t^2}{t^4+1} \, dt.$$ | [
"\\dfrac{\\pi - 2\\ln(1+\\sqrt{2})}{4\\sqrt{2}}"
] | [
"Calculate the integral $$\\int^{1}_{0} \\frac{t^2}{t^4+1} \\, dt.$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c9e4a5a2d0b8f24302cd37051258690b0041f6ed059f996925116510ecacad45 | Calculate the integral $$\int^{1}_{0}\frac{\sqrt{2}x(1-x)}{\sqrt{2}+2^x}dx.$$ | [
"\\(\\frac{1}{12}\\)"
] | [
"Calculate the integral $$\\int^{1}_{0}\\frac{\\sqrt{2}x(1-x)}{\\sqrt{2}+2^x}dx.$$"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7f3d87bf3b3ec3c6dbc75522c05fdc2c15cbfe986f09b2cc6dd8f6dd425773b2 | Calculate the integral $$\int_0^1 \frac{\arctan x \ln(1+x)}{x}dx.$$ | [
"\\(-\\dfrac{3}{32}\\pi\\ln^2 2 -\\dfrac{9}{128}\\pi^3 + \\dfrac{3}{2}\\texttt{G}\\ln 2 + 3\\Im \\operatorname{Li}_3\\left(\\dfrac{1+i}{2}\\right)\\)"
] | [
"Calculate the integral $$\\int_0^1 \\frac{\\arctan x \\ln(1+x)}{x}dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66e0f9db1bbb6c9a4f89c85c50d87ec2c7602e0f8daf101c7c8497b4d1ddf5a9 | Calculate the integral $$\int_0^1 \frac{\ln(1-x^2)\ln(1+x^2)}{1+x^2}dx.$$ | [
"\\(\\frac{\\pi^3}{32}-3G\\ln 2+\\frac{\\pi}{2}\\ln^22\\)"
] | [
"Calculate the integral\n$$\\int_0^1 \\frac{\\ln(1-x^2)\\ln(1+x^2)}{1+x^2}dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-01f474504ededc4bed87fe28d7bae11c35d47a386747540c1dfa2d7fda3da9bf | Calculate the integral $$\int_0^1 \frac{\ln^2 x \ln(1+x)}{1-x} \, dx.$$ | [
"\\(\\frac{7}{2}\\zeta(3)\\ln 2 - \\frac{19}{720}\\pi^4\\)"
] | [
"Calculate the integral $$\\int_0^1 \\frac{\\ln^2 x \\ln(1+x)}{1-x} \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-f9c37f54735fabb1faad8461a1590a735ad14695b861f144c62297f54b5901d8 | Calculate the integral $$\int_0^1 \frac{\ln^2 x}{\sqrt{2-x^2}}dx.$$ | [
"\\(-\\Im\\operatorname{Li}_3\\left(\\dfrac{1+i}{2}\\right)+\\dfrac{3}{32}\\pi\\ln^2 2 + \\dfrac{23}{384}\\pi^3\\)"
] | [
"Calculate the integral $$\\int_0^1 \\frac{\\ln^2 x}{\\sqrt{2-x^2}}dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9de0742f0a1b2a6632343d8721f814f47eb498148dad8cde687eeec4a0051d6d | Calculate the integral $$\int_0^1 \frac{\ln^3 x} {(1+x)^2} dx.$$ | [
"\\(-\\frac{9}{2}\\zeta(3)\\)"
] | [
"Calculate the integral $$\\int_0^1 \\frac{\\ln^3 x} {(1+x)^2} dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-84288d86155a5a25fa6724eb7363f11e20e1e0f1f1fa711c0ab651ba92280d61 | Calculate the integral $$\int_0^1 \frac{\ln^3(1+x)\ln(1-x)}{x}dx$$ | [
"\\(-6\\operatorname{Li}_5\\left(\\frac12\\right) - 6\\ln2\\operatorname{Li}_4\\left(\\frac12\\right) + \\frac{3}{4}\\zeta(5) + \\frac{21}{8}\\zeta(2)\\zeta(3) - \\frac{21}{8}\\ln^22\\zeta(3) + \\ln^32\\zeta(2) - \\frac15\\ln^52\\)"
] | [
"Calculate the integral\n$$\\int_0^1 \\frac{\\ln^3(1+x)\\ln(1-x)}{x}dx$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-114096d92dc70ed3f60c094043c02e1fe02722c21311ee2c0e57959c36e5a891 | Calculate the integral $$\int_0^1 \frac{x^3-1}{x+1}\frac{dx}{\ln x}.$$ | [
"\\(\\ln\\left(\\frac{3\\pi}{4}\\right)\\)"
] | [
"Calculate the integral $$\\int_0^1 \\frac{x^3-1}{x+1}\\frac{dx}{\\ln x}.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 4
} | 17 | 0 | 0 |
omr-ws-0ff91843958e2382f04499becfd7bdea59882662dd388cf6196eadfa644d4485 | Calculate the integral $$\int_0^1 \frac{x}{1-x} \ln x \ln(1-x) \, dx.$$ | [
"\\(\\zeta(3) + \\frac{\\pi^2}{6} - 2\\)"
] | [
"Calculate the integral $$\\int_0^1 \\frac{x}{1-x} \\ln x \\ln(1-x) \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 27
} | 46 | 0 | 0 |
omr-ws-b567af27a5ca330fb5d62d639cf559582f64fdfc928ac5a0642295fb2169b9c7 | Calculate the integral $$\int_0^1 \left(\dfrac{x}{2-x}\right)^{1/3}\text{d}x.$$ | [
"\\(\\frac{2\\pi}{3\\sqrt{3}} + \\frac{2\\ln 2}{3} - 1\\)"
] | [
"Calculate the integral $$\\int_0^1 \\left(\\dfrac{x}{2-x}\\right)^{1/3}\\text{d}x.$$"
] | [
"aops_c7_college_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-5202078c08f9d51ee2aca2c550120be3eaf3d2608f6e55da198d789463b61a38 | Calculate the integral $$\int_0^1 \log(x) \log(1-x) \, dx.$$ | [
"\\( 2 - \\frac{\\pi^2}{6} \\)"
] | [
"Calculate the integral $$\\int_0^1 \\log(x) \\log(1-x) \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-258a14f3c41044bd303ce5d44908bf44eb2a72156224e9165845c76ab9e01d26 | Calculate the integral $$\int_0^1 x^n \ln\left(\sqrt{1+x^k} - \sqrt{1-x^k}\right) \, dx$$ where \( k \geq 0 \) and \( n \geq 0 \) are real numbers. | [
"\\(\\frac{\\ln 2}{2n+2}-\\frac{k}{2(n+1)^2}-\\frac{\\sqrt{\\pi}\\Gamma(\\frac{n+1}{2k})}{4(k+1)\\Gamma(\\frac{n+k+1}{2k})}\\)"
] | [
"Calculate the integral $$\\int_0^1 x^n \\ln\\left(\\sqrt{1+x^k} - \\sqrt{1-x^k}\\right) \\, dx$$ where \\( k \\geq 0 \\) and \\( n \\geq 0 \\) are real numbers."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4dbe6c558a37a21bfdc87a709a2f9480e2fedf8eb0aabac943501e167e799d3 | Calculate the integral $$\int_0^1[nx]\frac{\ln x+\ln (1-x)}{\sqrt{x(1-x)}}dx$$ where $n$ is a given positive integer and $[a]$ represents the largest integer that does not exceed $a$. | [
"\\(-2(n-1)\\pi \\ln 2\\)."
] | [
"Calculate the integral\n$$\\int_0^1[nx]\\frac{\\ln x+\\ln (1-x)}{\\sqrt{x(1-x)}}dx$$\nwhere $n$ is a given positive integer and $[a]$ represents the largest integer that does not exceed $a$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-d333f955640212ceb82b99c5e790d3eb8e2e4689735b79e53070da403a4d61ec | Calculate the integral $$\int_0^1\frac{1}{1+\phi x^4}\cdot\frac{dx}{\sqrt{1-x^2}}$$ where $\phi$ is the golden ratio ($\phi = \frac{1+\sqrt{5}}{2}$). | [
"\\dfrac{\\pi(\\sqrt{5} - 1)}{4}"
] | [
"Calculate the integral\n$$\\int_0^1\\frac{1}{1+\\phi x^4}\\cdot\\frac{dx}{\\sqrt{1-x^2}}$$\nwhere $\\phi$ is the golden ratio ($\\phi = \\frac{1+\\sqrt{5}}{2}$)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f4844d1f18d4ac1eee982bf8b21a1db5c80f7c9105b595a2bfdf3042da65db2 | Calculate the integral $$\int_0^1\frac{\ln^2(1+x+x^2)}{x}dx.$$ | [
"\\(\\frac{2\\pi}{9\\sqrt3}\\psi_1\\left(\\frac13\\right)-\\frac{4\\pi^3}{27\\sqrt3}-\\frac23\\zeta(3)\\)"
] | [
"Calculate the integral $$\\int_0^1\\frac{\\ln^2(1+x+x^2)}{x}dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6503f385f2185ab51c4fdabc69d6fa8a804da256b28b387409785c4389da0bcd | Calculate the integral $$\int_0^1\frac{x \cos x}{\sqrt{1+x^2}}\mathrm{d}x.$$ | [
"\\sqrt{2} \\sin(1) - 1"
] | [
"Calculate the integral\n$$\\int_0^1\\frac{x \\cos x}{\\sqrt{1+x^2}}\\mathrm{d}x.$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c28ddae8f178f194bf0b14925c158f1eb84d40d62b095d33e5b93b0e3141c807 | Calculate the integral $$\int_0^1{x\ln ^2\left( \frac{x}{1+\sqrt{1-x^2}} \right)}\text{d}x.$$ | [
"\\ln 2"
] | [
"Calculate the integral $$\\int_0^1{x\\ln ^2\\left( \\frac{x}{1+\\sqrt{1-x^2}} \\right)}\\text{d}x.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-7a9864f228b1a6b36e83dbc15745830a6c7ab679858a087053c62028556e5541 | Calculate the integral $$\int_0^\frac{\pi}{2} \arcsin\left(\frac{\sin x}{2}\right)dx.$$ | [
"\\(\\frac{\\pi^2}{24}-\\frac14\\log^2(2)-\\frac12\\operatorname{Li}_2\\left(-\\frac12\\right)\\)"
] | [
"Calculate the integral\n$$\\int_0^\\frac{\\pi}{2} \\arcsin\\left(\\frac{\\sin x}{2}\\right)dx.$$"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e08cde554617f9bd094d0478bd68e1c291fe9839677d92b0216ea1dfe1e7d910 | Calculate the integral $$\int_0^\frac{\pi}{2} \arctan\left(\frac34\sin(2x)\right) \frac{dx} {\sin x}.$$ | [
"\\(\\frac{\\pi}{3}\\ln(2+\\sqrt{3})\\)"
] | [
"Calculate the integral $$\\int_0^\\frac{\\pi}{2} \\arctan\\left(\\frac34\\sin(2x)\\right) \\frac{dx} {\\sin x}.$$"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d953378f55667f1051ecf79511bac438ecca5ba7134471b3fd870aa4e0a00d89 | Calculate the integral $$\int_0^\frac{\pi}{2} \frac{x^2}{\sqrt{\cos x}} \, dx.$$ | [
"\\frac{\\sqrt{\\pi} \\, \\Gamma\\left(\\frac{1}{4}\\right)^2}{4 \\sqrt{2}}"
] | [
"Calculate the integral $$\\int_0^\\frac{\\pi}{2} \\frac{x^2}{\\sqrt{\\cos x}} \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-d7957be5d045a7e434fad1f1fd37b16d54323f5199ba31dea1eb631636cddf37 | Calculate the integral $$\int_0^\frac{\pi}{2} \frac{x^2}{\sqrt{\sin x}} \, dx.$$ | [
"\\frac{\\sqrt{\\pi} \\, \\Gamma\\left(\\frac{1}{4}\\right)^2}{4 \\sqrt{2}}"
] | [
"Calculate the integral $$\\int_0^\\frac{\\pi}{2} \\frac{x^2}{\\sqrt{\\sin x}} \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-514eafb58085c057d2be74ccac236c865ff7ad6756d38609e507e213ed288a33 | Calculate the integral $$\int_0^\frac{\pi}{2} \operatorname{arcsinh} (\sin x+\cos x) \, dx.$$ | [
"\\dfrac{\\pi}{2} \\ln(1 + \\sqrt{2})"
] | [
"Calculate the integral $$\\int_0^\\frac{\\pi}{2} \\operatorname{arcsinh} (\\sin x+\\cos x) \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-9dbafa23205fa9ef96371158324daccbfb4aee7556e2999805ba469959e813c4 | Calculate the integral $$\int_0^\frac{\pi}{2} x\ln^2 (\sin x)dx.$$ | [
"\\(-\\frac{19 \\pi^4}{2880}+\\frac{\\pi^2}{12}\\ln^22+\\frac{1}{24}\\ln^42+\\text{Li}_4(1/2)\\)"
] | [
"Calculate the integral $$\\int_0^\\frac{\\pi}{2} x\\ln^2 (\\sin x)dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a6ab90ad64f46947ae84a062018beacaf8c79222f46a88da41f999764df95b7 | Calculate the integral $$\int_0^\frac{\pi}{2} x^2 \sqrt{\cot x} \, dx.$$ | [
"\\(\\frac{\\sqrt{2}\\pi(5\\pi^2-12\\pi \\ln 2-12\\ln^22)}{96}\\)"
] | [
"Calculate the integral $$\\int_0^\\frac{\\pi}{2} x^2 \\sqrt{\\cot x} \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7db2b2fbff8cc5bb7bab62d0bb0afe81aa77e946a489803851f80dfaf18a9e8d | Calculate the integral $$\int_0^\frac{\pi}{2} x^2 \sqrt{\tan x} \, dx.$$ | [
"\\(\\frac{\\sqrt{2}\\pi(5\\pi^2+12\\pi\\ln 2 - 12\\ln^22)}{96}\\)"
] | [
"Calculate the integral $$\\int_0^\\frac{\\pi}{2} x^2 \\sqrt{\\tan x} \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f5bba98b9e0f648f39490486c636e3d93fe7817ebeee37c583ee010513c63f1 | Calculate the integral $$\int_0^\frac{\pi}{4}\ln{\sin{x}}\ln{\cos{x}}\,dx.$$ | [
"\\(\\frac{\\pi}{4} \\left( (\\ln 2)^2 - \\frac{\\pi^2}{24} \\right)\\)"
] | [
"Calculate the integral\n$$\\int_0^\\frac{\\pi}{4}\\ln{\\sin{x}}\\ln{\\cos{x}}\\,dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 8
} | 27 | 0 | 0 |
omr-ws-8a2a617a21ffc8528f556cb8883008f3e549e0d6e68a6b2ce50966a731643d94 | Calculate the integral $$\int_0^\infty \frac{(x^4-2)x^2}{\cosh(x\frac{\pi}2)}\,dx.$$ | [
"59"
] | [
"Calculate the integral $$\\int_0^\\infty \\frac{(x^4-2)x^2}{\\cosh(x\\frac{\\pi}2)}\\,dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 11
} | 29 | 0 | 0 |
omr-ws-d942977ca64ff67921448778a317dbf0fbb4241da24d4bcc0bf0ef9de0c9f462 | Calculate the integral $$\int_0^\infty\left(\frac{\cos ax-\cos bx}{x}\right)^2dx$$ where \(a, b \in \mathbb{R}\). | [
"\\(\\frac{\\pi}{2} |a-b|\\)"
] | [
"Calculate the integral $$\\int_0^\\infty\\left(\\frac{\\cos ax-\\cos bx}{x}\\right)^2dx$$ where \\(a, b \\in \\mathbb{R}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 12
} | 24 | 0 | 0 |
omr-ws-b4cac29d8aa2a39df988043e200b0269e39088593cf52f28ab5cb1c88b379dff | Calculate the integral $$\int_0^a \arctan(e^x-1)\,dx$$ where \(a\) is a positive number. | [
"\\( \\frac{1}{12} \\left( -22 a^3 \\arctan(1 - e^a) - 11 i a^3 \\log\\left(1 - \\left(\\frac{1}{2} + \\frac{i}{2}\\right) e^a\\right) + 11 i a^3 \\log\\left(1 - \\left(\\frac{1}{2} - \\frac{i}{2}\\right) e^a\\right) + 18 i a^2 \\text{PolyLog}\\left[2, \\left(\\frac{1}{2} - \\frac{i}{2}\\right) e^a\\right] - 18 i a... | [
"Calculate the integral $$\\int_0^a \\arctan(e^x-1)\\,dx$$ where \\(a\\) is a positive number."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b37e2902b579f1ca6dc15a3ba9d3ff4ed6410ce353a88e9728398c860ab44a8b | Calculate the integral $$\int_0^a x \arctan(e^x-1)\,dx$$ where \(a\) is a positive number. | [
"\\( \\frac{1}{12} \\left( -22 a^3 \\arctan(1 - e^a) - 11 i a^3 \\log\\left(1 - \\left(\\frac{1}{2} + \\frac{i}{2}\\right) e^a\\right) + 11 i a^3 \\log\\left(1 - \\left(\\frac{1}{2} - \\frac{i}{2}\\right) e^a\\right) + 18 i a^2 \\text{PolyLog}\\left[2, \\left(\\frac{1}{2} - \\frac{i}{2}\\right) e^a\\right] - 18 i a... | [
"Calculate the integral $$\\int_0^a x \\arctan(e^x-1)\\,dx$$ where \\(a\\) is a positive number."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1a80d3edc58441f951969f6ffd676acf0658811bfbf26e6d773b1a2286aed47 | Calculate the integral $$\int_0^a x^2 \arctan(e^x-1)\,dx$$ where \(a\) is a positive number. | [
"\\( \\frac{1}{12} \\left( -22 a^3 \\arctan(1 - e^a) - 11 i a^3 \\log\\left(1 - \\left(\\frac{1}{2} + \\frac{i}{2}\\right) e^a\\right) + 11 i a^3 \\log\\left(1 - \\left(\\frac{1}{2} - \\frac{i}{2}\\right) e^a\\right) + 18 i a^2 \\text{PolyLog}\\left[2, \\left(\\frac{1}{2} - \\frac{i}{2}\\right) e^a\\right] - 18 i a... | [
"Calculate the integral $$\\int_0^a x^2 \\arctan(e^x-1)\\,dx$$ where \\(a\\) is a positive number."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-3dbdb9ecaa92b72cecf0c07f5dc3e164dff14bff439ca2ada782ab37773e6180 | Calculate the integral $$\int_0^{+\infty} \frac{e^{-x^2}}{\left(x^2+\frac{1}{2}\right)^2}\; \text dx.$$ | [
"\\(\\sqrt{\\pi}\\)"
] | [
"Calculate the integral $$\\int_0^{+\\infty} \\frac{e^{-x^2}}{\\left(x^2+\\frac{1}{2}\\right)^2}\\; \\text dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 23
} | 41 | 0 | 0 |
omr-ws-e03ef1fdea36217db863baa26cd169bab8b31dddebe78738e23e5aac66f0e796 | Calculate the integral $$\int_0^{2\pi} \frac{\cos^2{3\theta}}{5-4\cos{2\theta}} d\theta$$ | [
"\\(\\frac{3\\pi}{8}\\)"
] | [
"Calculate the integral\n\n$$\\int_0^{2\\pi} \\frac{\\cos^2{3\\theta}}{5-4\\cos{2\\theta}} d\\theta$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 16
} | 24 | 0 | 0 |
omr-ws-9d5f85e434a9cae842e63bb22066977784a8cbf5f669fa125deec06b9d9bd533 | Calculate the integral $$\int_0^{\frac{\pi}{2}}\cos\left(2\pi\sin^2x\right)dx$$. | [
"-\\frac{\\pi}{2} J_0(\\pi)"
] | [
"Calculate the integral $$\\int_0^{\\frac{\\pi}{2}}\\cos\\left(2\\pi\\sin^2x\\right)dx$$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-0622057afa087e75eb0ae7c7b687f0b947d5eaf97acee226c983bdd4258a3763 | Calculate the integral $$\int_0^{\frac{\pi}{2}}\cos\left(\pi\sin x\right)dx$$. | [
"\\(\\frac{\\pi J_0(\\pi)}{2}\\)"
] | [
"Calculate the integral $$\\int_0^{\\frac{\\pi}{2}}\\cos\\left(\\pi\\sin x\\right)dx$$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-f428d2ee77710d5e221577ce7d7f9f7f4c35aac57f5a9ad783692e95aebc6c60 | Calculate the integral $$\int_0^{\frac{\pi}{4}} \frac{x^2}{(\cos x + x \sin x)^2} \, dx.$$ | [
"\\(\\frac{4-\\pi}{4+\\pi}\\)"
] | [
"Calculate the integral\n$$\\int_0^{\\frac{\\pi}{4}} \\frac{x^2}{(\\cos x + x \\sin x)^2} \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-bbe5c079f7abccf14e814a787e022db82a4e3adb7c54f6ca55acfc86ed9c4960 | Calculate the integral $$\int_0^{\infty} \frac{\{e^x\}}{e^x} \, dx$$ where $\{x\}$ denotes the fractional part of the number $x$. | [
"\\(1 - \\gamma\\) where \\(\\gamma\\) is the Euler-Mascheroni constant."
] | [
"Calculate the integral\n$$\\int_0^{\\infty} \\frac{\\{e^x\\}}{e^x} \\, dx$$\nwhere $\\{x\\}$ denotes the fractional part of the number $x$."
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-fe24431952d300756261a9f10e1561a391e5838bd16e9e7be4b8fcd9390fd1e0 | Calculate the integral $$\int_0^{\infty} x^{a-1} \sin\left(\frac{a\pi}{2} - bx\right) \frac{1}{x^2 - r^2} \, dx$$ given that \( r > 0 \), \( b > 0 \), and \( 0 < a < 2 \). | [
"\\frac{\\pi}{2} r^{a-2} \\frac{\\sin\\left(\\frac{a\\pi}{2} - b r\\right)}{\\sin\\left(\\frac{a\\pi}{2}\\right)}"
] | [
"Calculate the integral $$\\int_0^{\\infty} x^{a-1} \\sin\\left(\\frac{a\\pi}{2} - bx\\right) \\frac{1}{x^2 - r^2} \\, dx$$ given that \\( r > 0 \\), \\( b > 0 \\), and \\( 0 < a < 2 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0dffdc35ac6b7827cb39f49edd2a5928641e901e954b7db39f6b421848d2111 | Calculate the integral $$\int_0^{\pi/2} \log(\sin x) \, dx.$$ | [
"\\(-\\frac{\\pi}{2} \\ln 2\\)"
] | [
"Calculate the integral $$\\int_0^{\\pi/2} \\log(\\sin x) \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-97b835f66a2c00ddc2d2fb8b2e1595eefcf390e8b0d62f2c682d49affd19bf7d | Calculate the integral $$\int_0^{\pi/2} \sqrt{\frac{1}{\sin{2\theta}}} \, d\theta.$$ | [
"\\(\\frac{\\Gamma^2(1/4)}{2\\sqrt{2\\pi}}\\)"
] | [
"Calculate the integral\n$$\\int_0^{\\pi/2} \\sqrt{\\frac{1}{\\sin{2\\theta}}} \\, d\\theta.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 22
} | 36 | 0 | 0 |
omr-ws-5239d8bb72bf3997bc05e2772acbadb7f783ebf65c1cbb84cfe354663af9f3f6 | Calculate the integral $$\int_0^{\pi}{x^2\ln \left( a+\cos x \right) \text{d}x}$$ | [
"\\(\\frac{\\pi^3}{3}\\ln\\left(\\frac{a+\\sqrt{a^2-1}}{2}\\right)-4\\pi\\text{Li}_3(a-\\sqrt{a^2-1})\\)"
] | [
"Calculate the integral\n$$\\int_0^{\\pi}{x^2\\ln \\left( a+\\cos x \\right) \\text{d}x}$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-82f30c221c2d078976d67861d91096d2333ef54d1d56f5a9a5866eb20c4d840a | Calculate the integral $$\int_0^{\tfrac{\pi}{4}} \dfrac{x\ln\left(\tfrac{\cos x+\sin x}{\cos x-\sin x}\right)}{\cos x(\cos x+\sin x)}dx$$ | [
"\\(\\dfrac{G\\ln 2}{2}+\\dfrac{\\pi^3}{192}\\)"
] | [
"Calculate the integral\n$$\\int_0^{\\tfrac{\\pi}{4}} \\dfrac{x\\ln\\left(\\tfrac{\\cos x+\\sin x}{\\cos x-\\sin x}\\right)}{\\cos x(\\cos x+\\sin x)}dx$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f0903e5f5730ecad73b9769085552f3dcc734f5c21b2b77a71c543b17bd0b2e | Calculate the integral $$\int_L |y|\,ds$$ where $L$ is the curve given by the equation $r = a(1 + \cos\varphi)$. | [
"\\(\\frac{32a^2}{5}\\)"
] | [
"Calculate the integral\n$$\\int_L |y|\\,ds$$\nwhere $L$ is the curve given by the equation $r = a(1 + \\cos\\varphi)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-717f5bf9007ecb4a1cad66a599145ad863e3b9107a9cd74496d221ee4f59f793 | Calculate the integral $$\int_{-\ln2}^{0} \frac {1}{\sqrt{1-e^{2x}}} dx$$ | [
"\\(\\frac{\\ln(2+\\sqrt{3})}{2}\\)"
] | [
"Calculate the integral\n\n$$\\int_{-\\ln2}^{0} \\frac {1}{\\sqrt{1-e^{2x}}} dx$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7312aba479aba9ed3989f0b798e426ed13fb35f76958dfbab630a37ca5cc9ec7 | Calculate the integral $$\int_{-\sqrt{2}}^{0} \sin^{-1}{e^x} dx.$$ | [
"\\(\\frac{\\pi}{2} \\ln 2 - a \\ln (2 \\sin a) - \\frac{1}{2} \\texttt{Cl}_2(2a)\\)"
] | [
"Calculate the integral $$\\int_{-\\sqrt{2}}^{0} \\sin^{-1}{e^x} dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cebd3229ea2698ecdbec8a106209014b36f7cbf41c5421b034b6db249e1936a1 | Calculate the integral $$\int_{0}^{+\infty}\frac{e^{-ax}+e^{-bx}}{\cosh\left( \frac{x}{a}\right)}\cdot \frac{\tanh(cx)}{x}dx,$$ where \(a, b, c \in \mathbb{R}\) and \(a \neq 0\). | [
"\\ln\\left(\\frac{\\cosh(\\pi a c)}{\\cosh(\\pi b c)}\\right)"
] | [
"Calculate the integral\n$$\\int_{0}^{+\\infty}\\frac{e^{-ax}+e^{-bx}}{\\cosh\\left( \\frac{x}{a}\\right)}\\cdot \\frac{\\tanh(cx)}{x}dx,$$\nwhere \\(a, b, c \\in \\mathbb{R}\\) and \\(a \\neq 0\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0cd91d302e4f9e5b0ab37aa3ef8d64a1ddd26ec2183f2a0d01a7830d98460a79 | Calculate the integral $$\int_{0}^{+\infty}\frac{x^{-p}}{x^2 + 2x \cos(r) + 1}dx,$$ where $-1 < p < 1$ and $-\pi < r < \pi$. | [
"\\(\\frac{\\pi}{\\sin(\\pi p)} \\frac{\\sin(pr)}{\\sin(r)}\\)"
] | [
"Calculate the integral $$\\int_{0}^{+\\infty}\\frac{x^{-p}}{x^2 + 2x \\cos(r) + 1}dx,$$ where $-1 < p < 1$ and $-\\pi < r < \\pi$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-62890910576ba90f6a95c63f937396a16a881658493be1ea9924dd41e19744a0 | Calculate the integral $$\int_{0}^{1} \frac{\arctan \sqrt{2-x^{2}}}{1+x^{2}} \, \mathrm{d} x.$$ | [
"\\(\\frac{7\\pi^2}{96}\\)"
] | [
"Calculate the integral $$\\int_{0}^{1} \\frac{\\arctan \\sqrt{2-x^{2}}}{1+x^{2}} \\, \\mathrm{d} x.$$"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-b530a1d0208595434410f2608b6d59fa0b4853936328528f9d808d295bdb4f5d | Calculate the integral $$\int_{0}^{1} \frac{\ln(1-x)\ln(1+x)}{x} \, dx.$$ | [
"\\(-\\frac{\\pi^2}{12} + \\frac{(\\ln 2)^2}{2}\\)"
] | [
"Calculate the integral\n$$\\int_{0}^{1} \\frac{\\ln(1-x)\\ln(1+x)}{x} \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16bc81138f4ecad5972871e05cba85534058bce648a80324841a152bb03abccd | Calculate the integral $$\int_{0}^{1} \frac{\log_{e}\left( x+\sqrt{1-x^{2}}\right)}{\sqrt{1+x^{2}}}dx.$$ | [
"\\frac{1}{2} \\left( \\ln(1 + \\sqrt{2}) \\right)^2"
] | [
"Calculate the integral $$\\int_{0}^{1} \\frac{\\log_{e}\\left( x+\\sqrt{1-x^{2}}\\right)}{\\sqrt{1+x^{2}}}dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-4b23d9bcf4877b831d0c1c9a6ec800dee867bf5817092d34813e23ad580307d4 | Calculate the integral $$\int_{0}^{1} \sqrt{x^{n}+\sqrt{x^{n}+\sqrt{x^{n}+\sqrt{x^{n}+\cdots}}}} \, dx,$$ where \( n \in \mathbb{N} \). | [
"\\frac{1}{2} \\left(1 + {}_2F_1\\left(-\\frac{1}{2}, \\frac{1}{n}; 1 + \\frac{1}{n}; -4\\right)\\right)"
] | [
"Calculate the integral $$\\int_{0}^{1} \\sqrt{x^{n}+\\sqrt{x^{n}+\\sqrt{x^{n}+\\sqrt{x^{n}+\\cdots}}}} \\, dx,$$ where \\( n \\in \\mathbb{N} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-01b81b13bf37eda3a4b34678a19f85342301996aa42490fc5c186a406d6cdfa7 | Calculate the integral $$\int_{0}^{1} x (\tan^{-1}{x})^2 dx.$$ | [
"\\(\\frac{\\pi^2}{16} - \\frac{\\pi}{4} + \\frac{\\ln{2}}{2}\\)"
] | [
"Calculate the integral $$\\int_{0}^{1} x (\\tan^{-1}{x})^2 dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 30
} | 53 | 0 | 0 |
omr-ws-785287f8ff66eb0515ce7942bc10d886cfd02b61d5fac4d3e31970501ce19ac5 | Calculate the integral $$\int_{0}^{1}\cos \left (\ln \frac{1}{x} \right )\frac{x^b-x^a}{\ln x}dx$$ | [
"\\(\\frac{1}{2}\\ln\\frac{(b+1)^2+1}{(a+1)^2+1}\\)"
] | [
"Calculate the integral\n$$\\int_{0}^{1}\\cos \\left (\\ln \\frac{1}{x} \\right )\\frac{x^b-x^a}{\\ln x}dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 22
} | 36 | 0 | 0 |
omr-ws-b64ad6bfa539039f30977882794d1dab5275b88a18768d151a897af1b4794a21 | Calculate the integral $$\int_{0}^{1}\frac{\ln x \ln(1+x)}{1-x} \, dx$$ | [
"\\frac{\\pi^2}{2} \\ln 2 - \\frac{7}{4} \\zeta(3)"
] | [
"Calculate the integral\n$$\\int_{0}^{1}\\frac{\\ln x \\ln(1+x)}{1-x} \\, dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-20a23f6e5841bd573d8fd8bfcf639fe6d2d7a5844b889b0f3297cca661a63f52 | Calculate the integral $$\int_{0}^{1}\frac{\ln(1-x)}{1+x}dx$$ | [
"\\(\\frac{\\ln^2 2}{2} - \\frac{\\pi^2}{12}\\)"
] | [
"Calculate the integral\n$$\\int_{0}^{1}\\frac{\\ln(1-x)}{1+x}dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 27
} | 48 | 0 | 0 |
omr-ws-87eadd179342350a8d8c0aa1e5ccac6fa23fd23a2d88ae71678e3c8a016abcc4 | Calculate the integral $$\int_{0}^{1}\frac{\ln(1-x^2)}{x^2}dx$$ | [
"-2\\ln 2"
] | [
"Calculate the integral\n$$\\int_{0}^{1}\\frac{\\ln(1-x^2)}{x^2}dx$$"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-b42b9c1c59ed614836f8d5e3b194395948a35c2ba857017dd043a01ae83f7ff2 | Calculate the integral $$\int_{0}^{1}\frac{\ln^2(1-x^2)}{x^2}dx$$ | [
"\\(\\frac{\\pi^2}{3} - 4\\ln^2{2}\\)"
] | [
"Calculate the integral\n$$\\int_{0}^{1}\\frac{\\ln^2(1-x^2)}{x^2}dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-1c8547216d0ff749d9d23aaeb750ce65543f840bba26f691fe0a2fd10aa99676 | Calculate the integral $$\int_{0}^{1}x^{3}\ln^{3}\left( \frac{1}{x}\right)dx.$$ | [
"\\(\\frac{3}{128}\\)"
] | [
"Calculate the integral $$\\int_{0}^{1}x^{3}\\ln^{3}\\left( \\frac{1}{x}\\right)dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-5bee1e70a53d11329a811b15b9db2431e75575966836ebde55c28c0803afc3d0 | Calculate the integral $$\int_{0}^{2\pi} \prod_{k=1}^{n}\cos(kx)dx.$$ | [
"0"
] | [
"Calculate the integral $$\\int_{0}^{2\\pi} \\prod_{k=1}^{n}\\cos(kx)dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1bc9dd0303e93a4503ea9a31913511e8324b6c26b967eadde356eccb1e90520a | Calculate the integral $$\int_{0}^{2\pi} \prod_{k=1}^{n}\sin(kx)dx.$$ | [
"0"
] | [
"Calculate the integral $$\\int_{0}^{2\\pi} \\prod_{k=1}^{n}\\sin(kx)dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-ccc0e702a5166b942c80c73564dfb0c82fd17c864649f019faeefaa4223a2e69 | Calculate the integral $$\int_{0}^{\frac{\pi }{2}}\ln(2\cos x)dx$$ | [
"0"
] | [
"Calculate the integral\n$$\\int_{0}^{\\frac{\\pi }{2}}\\ln(2\\cos x)dx$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-afd0b41240adcbf339ea072b304eaf5e46a919f364fb7acbdd8c2b8feb90c6f7 | Calculate the integral $$\int_{0}^{\frac{\pi}{2}}\frac{\{tanx\}}{tanx}dx$$ where $\{u\}$ denotes the fractional part of $u$, i.e., $u - \lfloor u \rfloor$. | [
"\\(\\frac{\\pi}{2}+\\log\\left(\\prod_{k=1}^{\\infty}\\frac{k}{\\sqrt{k^2 + 1}}\\right)\\)"
] | [
"Calculate the integral\n\n$$\\int_{0}^{\\frac{\\pi}{2}}\\frac{\\{tanx\\}}{tanx}dx$$\n\nwhere $\\{u\\}$ denotes the fractional part of $u$, i.e., $u - \\lfloor u \\rfloor$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cd9088a3b59dcdf4a35fe28fce49d6a1f0b31f977f8068643825fbdda7148624 | Calculate the integral $$\int_{0}^{\frac{\pi}{3}}\frac{\cos(x)-\sin(x)}{\sin(x)+\cos(x)}\ln\left( \sin\left(\frac{x}{2} \right)\right)dx.$$ | [
"-\\frac{1}{2} (\\ln 2)^2"
] | [
"Calculate the integral $$\\int_{0}^{\\frac{\\pi}{3}}\\frac{\\cos(x)-\\sin(x)}{\\sin(x)+\\cos(x)}\\ln\\left( \\sin\\left(\\frac{x}{2} \\right)\\right)dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-2df5fdbf15cbc54548543815f2aaf2c418ef0aae0e4f1312eca44a90e1ba32e8 | Calculate the integral $$\int_{0}^{\infty }\frac{\cos mx}{1-2a\cos x+a^2}dx, \; |a|<1, \; m\in \mathbb{N}$$ | [
"\\dfrac{\\pi a^{m}}{1 - a^{2}}"
] | [
"Calculate the integral\n$$\\int_{0}^{\\infty }\\frac{\\cos mx}{1-2a\\cos x+a^2}dx, \\; |a|<1, \\; m\\in \\mathbb{N}$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee9a575aa4c6e833d90c546f5d77b150cd75d1959a8eff0529b8ef00b0fbdce6 | Calculate the integral $$\int_{0}^{\infty} \arctan(x) \cdot \ln\left(1+\frac{1}{x^2}\right) \, dx.$$ | [
"\\(\\frac{\\pi^2}{6}\\)"
] | [
"Calculate the integral\n$$\\int_{0}^{\\infty} \\arctan(x) \\cdot \\ln\\left(1+\\frac{1}{x^2}\\right) \\, dx.$$"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 22
} | 26 | 0 | 0 |
omr-ws-2f2c9aac03820281c36aa1303b6fea57b59e1283b6d63bc719a6cdd1aa49d776 | Calculate the integral $$\int_{0}^{\infty} \text{Li}_{2}(e^{-\pi x}) \arctan(x) \, dx$$ | [
"\\dfrac{\\pi^3}{320}"
] | [
"Calculate the integral\n$$\\int_{0}^{\\infty} \\text{Li}_{2}(e^{-\\pi x}) \\arctan(x) \\, dx$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-e21db51691b09bb0cab12dea6e4db072ea84a941c1243922d26bb1f3d8c8753e | Calculate the integral $$\int_{0}^{\infty}\frac{(x-3)e^{-x}}{x^3(e^x-1)}dx$$ | [
"\\text{Divergent}"
] | [
"Calculate the integral\n$$\\int_{0}^{\\infty}\\frac{(x-3)e^{-x}}{x^3(e^x-1)}dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-285c696cb05fe0af5ab56901da0cfdbb3f492d4de79cd6de30950cbf3646573f | Calculate the integral $$\int_{0}^{\infty}\frac{\cos(ax)}{e^{x}+e^{-x}}dx$$ where \( a \in \mathbb{R} \). | [
"\\(\\frac{\\pi}{4} \\text{sech} \\left(\\frac{a \\pi}{2}\\right)\\)"
] | [
"Calculate the integral\n$$\\int_{0}^{\\infty}\\frac{\\cos(ax)}{e^{x}+e^{-x}}dx$$\nwhere \\( a \\in \\mathbb{R} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
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