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 51 Prove $\int_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$ 36 Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only 33 A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$ 28 How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$? 26 Integral $\int_0^{\infty} \frac{\log x}{\cosh^2x} \ \mathrm{d}x = \log\frac {\pi}4- \gamma$

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 +25 Problem over a definite integral, which surely needs contour integration +10 Prove that $\sum_{n=1}^{\infty }\frac{B_{2n}}{(2n-1)!}=\frac{1}{2}-\frac{1}{(e-1)^2}$ +10 convergence of a generalized Riemann integral +5 Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function

### Questions (85)

 58 Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function 32 Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}}$ 24 When can't a real definite integral be evaluated using contour integration? 21 Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform 20 Evaluating $\int_{0}^{\infty} \frac{x^{3}- \sin^{3}(x)}{x^{5}} \ dx$ using contour integration

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