17 Unexpected appearances of $\pi^2 /~6$. 11 How to integrate $\int_{0}^{2\pi} \frac{1}{\sin^4x + \cos^4 x} \,dx$ 9 solve:$\int \frac{e^x+1}{e^x-1}dx$ 7 Evaluating $\int\limits_0^\infty \mathcal{e}^{-x}\ln^{2}x\,dx$ 6 Computing the integral $x^2\textrm{sech}^2(x)$

### Reputation (12,708)

 +10 I need to develop $P_n(X)=\frac{(X+i)^{2n+1} - (X - i)^{2n+1}}{2i}$ using the binomial coefficients formula and show a few properties +10 Can the Fibonacci sum $\sum_{i=0}^N x^i F_{a+ib}$, for integers $x$, $a$, $b$, be determined by a direct formula or generating function? +10 solve:$\int \frac{e^x+1}{e^x-1}dx$ +10 Minimum value of $\left(\frac{a-b}{b-c}\right)^{4}+\left(\frac{b-c}{c-a}\right)^4+\left(\frac{c-a}{a-b}\right)^{4}$

### Questions (152)

 29 Freaky dots in the complex plane 20 A magnificent series for $\pi-333/106$ 19 $\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{ax^2+bx+c}=\pi$ similar identities 18 Relationship between Catalan's constant and $\pi$ 17 product= $\exp\left[\frac{47\mathrm G}{30\pi}+\frac34\right]\left(\frac{11^{11}3^3}{13^{13}}\right)^{1/20}\sqrt{\frac{3}{7^{7/6}\pi}\sqrt{\frac2\pi}}$

### Tags (166)

 286 integration × 274 39 riemann-zeta × 13 185 calculus × 160 34 trigonometry × 33 143 definite-integrals × 114 33 indefinite-integrals × 34 87 sequences-and-series × 110 31 big-list × 14 55 real-analysis × 57 29 improper-integrals × 19

### Bookmarks (437)

 656 Splitting a sandwich and not feeling deceived 464 “The Egg:” Bizarre behavior of the roots of a family of polynomials. 225 “Integral Milking” 188 Are $14$ and $21$ the only “interesting” numbers? 178 Meaning of Rays in Polar Plot of Prime Numbers