How to show that $\int_0^1x^{-x}dx = \sum_{n=1}^\infty n^{-n}$?
24508
Series as an integral (sophomore's dream)
21616
Calculate integral $\int_{-1}^{1} (1-x^2)^n dx $
140638
Integrating $\int_0^{\infty} \frac{dx}{1+x^3}$ using residues.
19514
$\int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2)$ How to show?
109901
Computing $\int_0^\infty\frac1{(x+1)(x+2)\cdots(x+n)}\mathrm dx $
57224
Compute $\int_0^\infty \frac{dx}{1+x^3}$
145183
finding $\int_0^\infty \dfrac{dx}{1+x^4}$ through complex analysis
8829
How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$
98933
Calculating the sum of $f(x) = \sum_{n=0}^{\infty} {n \cdot 2^n \cdot x^n}$
51284
$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus
72502
Evaluate $\sum_{n=1}^\infty nx^{n-1}$
138054
Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$
148613
Integrate $\int_{0}^{1}{x^{-x}(1-x)^{x-1}\sin{\pi x}dx}$
98344
Prove that $\int_0^1[f''(x)]^2dx\ge4.$
131876
Show that $\int_{a}^{b}{x^{n}f(x)dx}=0$, then $f=0$
98710
Finding $\int_0^{\infty}xe^{-\lambda x} \, dx$
51509
How to show that $1 \over \sqrt{1 - 4x} $ generates $\sum_{n=0}^\infty \binom{2n}{n}x^n $