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JamesJ
  • Member for 4 years, 10 months
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14 votes
6 answers
371 views

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

10 votes
1 answer
385 views

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

8 votes
1 answer
269 views

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

7 votes
3 answers
311 views

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

6 votes
2 answers
503 views

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

5 votes
1 answer
281 views

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

5 votes
2 answers
311 views

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

4 votes
1 answer
338 views

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

4 votes
1 answer
262 views

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

4 votes
0 answers
74 views

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

3 votes
3 answers
229 views

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

3 votes
2 answers
78 views

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

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]

2 votes
3 answers
167 views

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

2 votes
1 answer
67 views

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

2 votes
2 answers
254 views

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

2 votes
1 answer
115 views

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

2 votes
2 answers
126 views

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

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]

1 vote
1 answer
94 views

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

1 vote
2 answers
149 views

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

1 vote
2 answers
321 views

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

1 vote
2 answers
212 views

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