User JamesJ

14 votes

6 answers

371 views

How to evaluate the sum $\sum_{n=1}^{\infty}\frac{1}{(2n+1)(2n+2)}\left(1+\frac{1}{2}+...+\frac{1}{n}\right)$

asked Oct 20, 2017 at 14:39

10 votes

1 answer

385 views

How to evaluate the integral $\int_0^{\pi/2}x^2(\sin x+\cos x)^3\sqrt{\sin x\cos x} \, dx$?

calculus integration trigonometry closed-form

asked Jan 9, 2018 at 6:54

8 votes

1 answer

269 views

How to evaluate the integral $\int_0^1\frac{\arctan x}{x}\frac{1-x^3}{1+x^3}dx$

calculus integration definite-integrals logarithms trigonometric-integrals

asked Sep 17, 2017 at 8:29

7 votes

3 answers

311 views

Whats the result of double integral $\int_{0}^{1}\int_{0}^{1}\frac{1+x^2}{1+x^2+y^2}dxdy$

calculus integration analysis definite-integrals

asked Jan 11, 2018 at 10:47

6 votes

2 answers

503 views

How to evaluate $\int_0^1\ln (\sqrt{1-x}+\sqrt{x}+\sqrt{1+x})dx$

calculus integration definite-integrals logarithms

asked Sep 6, 2017 at 11:14

5 votes

1 answer

281 views

Is this true $n!\leq(\frac{5n+7}{12})^n,n∈N$?

inequality summation factorial radicals a.m.-g.m.-inequality

asked Aug 24, 2017 at 8:43

5 votes

2 answers

311 views

Ask a about hard integral of $\int_{0}^{\infty} \log x \log (\frac{a^2}{x^2}+1) \log(\frac{b^2}{x^2}+1)dx$

calculus real-analysis integration definite-integrals

asked Sep 21, 2018 at 10:13

4 votes

1 answer

338 views

How to find the closed form of the integral$\int_{0}^{\infty}f(x)\frac{\sin^nx}{x^m}dx$

calculus integration definite-integrals

asked Aug 26, 2017 at 6:41

4 votes

1 answer

262 views

How to evaluate $\int \frac{\arctan x}{x\sqrt {1+x^2}}\exp\left(-\frac{\arctan x}{x}\right)\,dx$

calculus integration algebra-precalculus indefinite-integrals

asked Sep 3, 2017 at 11:47

4 votes

0 answers

74 views

How to evaluate this $\int_{0}^{\pi/2}(\cos x)^{a+b}\cos {(b-a)x}\, dx$

calculus integration trigonometry

asked Jan 20, 2018 at 7:56

3 votes

3 answers

229 views

How do I find the closed form of this integral $\int_0^2\frac{\ln x}{x^3-2x+4}dx$?

calculus integration definite-integrals partial-fractions

asked Sep 8, 2017 at 5:45

3 votes

2 answers

78 views

How to find the limit:$\lim_{n\to \infty}\left(\sum_{k=n+1}^{2n}\left(2(2k)^{\frac{1}{2k}}-k^{\frac{1}{k}}\right)-n\right)$

calculus analysis limits summation

asked Jan 6, 2018 at 7:53

3 votes

1 answer

235 views

How to evaluate the integral $\int_{0}^{\pi}\arctan\left(\frac{1+\cos x}{2+\sin x}\right)dx$ [closed]

calculus integration inverse-function

asked Oct 19, 2017 at 19:40

2 votes

3 answers

167 views

Find $\int^\infty_0 \frac{\sin x}{x(1+\cos ^2 x)}dx$

calculus integration definite-integrals improper-integrals

asked Feb 5, 2021 at 6:03

2 votes

1 answer

67 views

How to solve this recurrence equation?$A_{n,k}=(n+1-k)A_{n-1,k-1}+(k+1)A_{n-1,k+1}$?

sequences-and-series trigonometry recurrence-relations combinations

asked Sep 5, 2017 at 9:24

2 votes

2 answers

254 views

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

calculus integration definite-integrals substitution

asked Sep 5, 2017 at 20:22

2 votes

1 answer

115 views

How to prove that $\int_0^1 \frac{x^{1-p}(1-x)^pdx}{x^2+1}=\frac{\pi}{\sin px}(2^{\frac{p}{2}}\cos {\frac{p\pi}{4}}-1)$

calculus integration definite-integrals beta-function

asked Jan 22, 2018 at 19:24

2 votes

2 answers

126 views

How to solve the Diophantine equation：$x^2+xy+y^2=r^2,x,y,r\in \Bbb Z$

number-theory elementary-number-theory diophantine-equations

asked Sep 26, 2017 at 9:41

2 votes

0 answers

116 views

How to prove that: $\lim_{n\to\infty} \frac{1^n+2^n+\cdots+n^n}{n^n} = \frac{e}{e-1}$ [duplicate]

calculus sequences-and-series limits summation

asked Sep 27, 2017 at 4:51

1 vote

1 answer

94 views

On the convergence of $\sqrt{1+\sqrt[3]{2+...+\sqrt[n+1]{n}}}$.

calculus real-analysis sequences-and-series limits

asked Oct 17, 2017 at 5:28

1 vote

2 answers

149 views

I want to find all the rational solutions of $x^2+y^3=z^3$

number-theory diophantine-equations

asked Sep 2, 2017 at 9:22

1 vote

2 answers

321 views

How to evaluate the integral $\int_0^2 \frac{\ln x}{\sqrt {x^2-2x+2}}dx$?

calculus integration definite-integrals logarithms polylogarithm

asked Sep 9, 2017 at 11:01

1 vote

2 answers

212 views

How to evaluate this $\int_{-1}^1 (\frac{1}{x}+1-\frac{\sqrt{1-x^2}}{x})\arctan\frac{2}{x^2}dx$

calculus integration definite-integrals inverse-function trigonometric-integrals

asked Sep 15, 2017 at 5:12

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23 Questions

How to evaluate the sum $\sum_{n=1}^{\infty}\frac{1}{(2n+1)(2n+2)}\left(1+\frac{1}{2}+...+\frac{1}{n}\right)$

How to evaluate the integral $\int_0^{\pi/2}x^2(\sin x+\cos x)^3\sqrt{\sin x\cos x} \, dx$?

How to evaluate the integral $\int_0^1\frac{\arctan x}{x}\frac{1-x^3}{1+x^3}dx$

Whats the result of double integral $\int_{0}^{1}\int_{0}^{1}\frac{1+x^2}{1+x^2+y^2}dxdy$

How to evaluate $\int_0^1\ln (\sqrt{1-x}+\sqrt{x}+\sqrt{1+x})dx$

Is this true $n!\leq(\frac{5n+7}{12})^n,n∈N$?

Ask a about hard integral of $\int_{0}^{\infty} \log x \log (\frac{a^2}{x^2}+1) \log(\frac{b^2}{x^2}+1)dx$

How to find the closed form of the integral$\int_{0}^{\infty}f(x)\frac{\sin^nx}{x^m}dx$

How to evaluate $\int \frac{\arctan x}{x\sqrt {1+x^2}}\exp\left(-\frac{\arctan x}{x}\right)\,dx$

How to evaluate this $\int_{0}^{\pi/2}(\cos x)^{a+b}\cos {(b-a)x}\, dx$

How do I find the closed form of this integral $\int_0^2\frac{\ln x}{x^3-2x+4}dx$?

How to find the limit:$\lim_{n\to \infty}\left(\sum_{k=n+1}^{2n}\left(2(2k)^{\frac{1}{2k}}-k^{\frac{1}{k}}\right)-n\right)$

How to evaluate the integral $\int_{0}^{\pi}\arctan\left(\frac{1+\cos x}{2+\sin x}\right)dx$ [closed]

Find $\int^\infty_0 \frac{\sin x}{x(1+\cos ^2 x)}dx$

How to solve this recurrence equation?$A_{n,k}=(n+1-k)A_{n-1,k-1}+(k+1)A_{n-1,k+1}$?

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

How to prove that $\int_0^1 \frac{x^{1-p}(1-x)^pdx}{x^2+1}=\frac{\pi}{\sin px}(2^{\frac{p}{2}}\cos {\frac{p\pi}{4}}-1)$

How to solve the Diophantine equation：$x^2+xy+y^2=r^2,x,y,r\in \Bbb Z$

How to prove that: $\lim_{n\to\infty} \frac{1^n+2^n+\cdots+n^n}{n^n} = \frac{e}{e-1}$ [duplicate]

On the convergence of $\sqrt{1+\sqrt[3]{2+...+\sqrt[n+1]{n}}}$.

I want to find all the rational solutions of $x^2+y^3=z^3$

How to evaluate the integral $\int_0^2 \frac{\ln x}{\sqrt {x^2-2x+2}}dx$?

How to evaluate this $\int_{-1}^1 (\frac{1}{x}+1-\frac{\sqrt{1-x^2}}{x})\arctan\frac{2}{x^2}dx$