# Random Variable

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 39 Prove $\int_{0}^{\frac{\pi}{2}}\frac{dx}{1+\sin^2{(\tan{x})}}=\dfrac{\pi}{2\sqrt{2}}\left(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\right)$ 31 Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only 26 About a harmonic series problem : How to prove that $\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$ 25 How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$? 21 Integral $\int_0^{\infty} \frac{\log x}{\cosh^2x} \ \mathrm{d}x = \log\frac {\pi}4- \gamma$

# 15,658 Reputation

 +10 Series expansion of $\coth x$ using the Fourier transform +10 Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$ +10 Compute $I=\int_0^{+\infty}\frac{\arctan(t)}{e^{\pi t}-1}dt$ +10 Calculate:$y'$ for $y = x^{x^{x^{x^{x^{.^{.^{.^{\infty}}}}}}}}$ and $y = \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+…\infty}}}}$

# 80 Questions

 25 Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}}$ 23 When can't a real definite integral be evaluated using contour integration? 22 The log gamma integral $\int_{0}^{z} \log \Gamma (x) \ \mathrm dx$ 18 Evaluating $\int_{0}^{\infty} \frac{x^{3}- \sin^{3}(x)}{x^{5}} \ dx$ using contour integration 17 Evaluating $\prod\limits_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right)$

# 73 Tags

 784 integration × 148 211 improper-integrals × 30 614 calculus × 82 164 sequences-and-series × 33 566 definite-integrals × 71 159 special-functions × 27 247 real-analysis × 38 146 contour-integration × 43 240 complex-analysis × 104 127 closed-form × 11

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 Mathematics 15,658 rep 12980