151 How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$ 77 Compute $\int_0^{\pi/4}\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)} x\exp(\frac{x^2-1}{x^2+1}) dx$ 52 Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^32^n}$ 50 How to find ${\large\int}_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$? 50 Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

### Reputation (24,601)

 +10 Cauchy distribution characteristic function +10 Deriving Mean and Variance of Laplace Distribution +10 Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$ +10 Evaluate $\int_0^\infty\frac{\ln x}{1+x^2}dx$

### Questions (7)

 29 Closed-forms for several tough integrals 25 An equation that generates a beautiful or unique shape for motivating students in mathematics 9 $1 + 1 + 1 +\cdots = -\frac{1}{2}$ 9 Random Triangle Inscribed in a Circular Sector 7 Dual Integral Problems: $\int\left(\pi^{\sin^2x}+e^{\sin^2x}\right)^2\cos2x~dx$ and $\int\frac{\sin x}{1+\cos x+e^x}dx$

### Tags (120)

 2k calculus × 255 393 real-analysis × 42 1k integration × 195 340 trigonometry × 56 1k definite-integrals × 121 326 closed-form × 13 645 improper-integrals × 64 311 indefinite-integrals × 60 396 algebra-precalculus × 76 266 sequences-and-series × 34

### Bookmarks (17)

 494 Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$ 173 Symmetry of function defined by integral 131 How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$ 113 Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$? 69 Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$