Kemono Chen

A Chinese high school student, interested in evaluating integrals and other mathematical things.

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Evaluating $\int_0^1\ln(1+x^2)\ln(x^2+x^3)\frac{dx}{1+x^2}$

calculus integration definite-integrals asked Oct 13 '18 at 13:23

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28 votes

Evaluate $\int_0^{\pi/4}{(4\cot x\ln\sec x-x)\ln^2\tan xdx}$

calculus integration definite-integrals asked Jul 12 '19 at 1:08

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22 votes

Evaluate $\lim\limits_{x\to\infty}\frac1x\int_0^x\max\{\sin t,\sin(t\sqrt2)\}dt$

calculus integration definite-integrals asked Jul 22 '18 at 9:12

math.stackexchange.com

21 votes

Evaluating $\int_0^1\frac{x^{2/3}(1-x)^{-1/3}}{1-x+x^2}dx$

calculus integration definite-integrals hypergeometric-function asked Feb 24 '19 at 11:37

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16 votes

Evaluate $\int_0^\infty{\frac{\tan x}{x^n}dx}$

calculus integration definite-integrals asked Mar 17 '18 at 22:33

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16 votes

Limit $\lim_{n\to\infty}\sum_{k=0}^n\frac{n^{2k}}{(k!)^2}\big/\sum_{k=0}^\infty\frac{n^{2k}}{(k!)^2}$

sequences-and-series limits summation asked Feb 20 '20 at 9:54

math.stackexchange.com

15 votes

On the relationship between $\Re\operatorname{Li}_n(1+i)$ and $\operatorname{Li}_n(1/2)$ when $n\ge5$

calculus definite-integrals summation polylogarithm asked Nov 23 '19 at 8:41

math.stackexchange.com

15 votes

Finding $\max |A|$ with $a_{ij}=\pm 1$

linear-algebra determinant asked Sep 10 '18 at 8:31

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14 votes

Entire function $f(z)$ grows like $\exp(x^\pi)$ as $x\to+\infty$

complex-analysis limits entire-functions asked Jan 15 '19 at 5:52

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13 votes

Evaluating $\int_0^{\pi/2}\operatorname{arcsinh}(2\tan x)\,dx$

calculus integration definite-integrals polylogarithm asked Jan 22 '19 at 8:15

math.stackexchange.com

Top Answers

Evaluate $\int_0^1 \log \left( \frac{x^2+\sqrt{3}x+1}{x^2-\sqrt{3}x+1} \right) \frac{dx}{x} $

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Evaluating $\sum_{n=1}^\infty\frac{(H_n)^2}{n}\frac{\binom{2n}n}{4^n}$

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A generalization for Serret's integral $\int_0^a \frac{\ln(1+ax)}{1+x^2}dx$

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Compute polylog of order $3$ at $\frac{1}{2}$

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How to prove $\int_0^\infty \ln(1+\frac{z}{\cosh(x)})dx=\frac{\pi^2}{8}+\frac{(\cosh^{-1}(z))^2}{2},z\ge1$ and a closed form for $-1<z<1$?

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Proving $\int\limits^{\infty}_{-\infty}\frac{\sin(2n\arctan(x))}{\left(x^2+1\right)^n\left( e^{x\pi}+1\right)}dx=\eta(2n)-\zeta(2n)$

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Solving $\int_{0}^{\infty} \frac{\sin(x)}{x^3}dx$

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How to compute the limit,$\lim_{x\rightarrow 0}\frac{3x^2-3x\sin x}{x^2+x\cos1/x}$

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Is $\sqrt{x}$ an even function

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Evaluating $\int_0^1\frac{\ln(1+x-x^2)}xdx$ without using polylogarithms.

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How to prove $\int_{-\infty}^{+\infty} \frac{x^2}{\cosh(x)^2} dx = \frac{\pi^2}{6}$?

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Kemono Chen

Evaluating $\int_0^1\ln(1+x^2)\ln(x^2+x^3)\frac{dx}{1+x^2}$

Evaluate $\int_0^{\pi/4}{(4\cot x\ln\sec x-x)\ln^2\tan xdx}$

Evaluate $\lim\limits_{x\to\infty}\frac1x\int_0^x\max\{\sin t,\sin(t\sqrt2)\}dt$

Evaluating $\int_0^1\frac{x^{2/3}(1-x)^{-1/3}}{1-x+x^2}dx$

Evaluate $\int_0^\infty{\frac{\tan x}{x^n}dx}$

Limit $\lim_{n\to\infty}\sum_{k=0}^n\frac{n^{2k}}{(k!)^2}\big/\sum_{k=0}^\infty\frac{n^{2k}}{(k!)^2}$

On the relationship between $\Re\operatorname{Li}_n(1+i)$ and $\operatorname{Li}_n(1/2)$ when $n\ge5$

Finding $\max |A|$ with $a_{ij}=\pm 1$

Entire function $f(z)$ grows like $\exp(x^\pi)$ as $x\to+\infty$

Evaluating $\int_0^{\pi/2}\operatorname{arcsinh}(2\tan x)\,dx$