Prove that $(n+1)\int_0^1{\frac{x^n}{x+1}dx}\to\frac12$
11489
Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$
157931
Contour integration of $\int \frac{dx} {(1+x^2)^{n+1}}$
132124
$\int_{0}^{\infty}\frac{x}{x^3+1}dx$ =?
37871
Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$
46359
Finding the general integrals of functions like $\frac1{x^n+1}$, $\cos^nx$.
72600
Proof that $n!\leq {(\frac{n+1}{2})}^{n}$
80144
Prove that $x-1$ divides $x^n-1$
117640
Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $
147977
Evaluate $\int_0^1\left(\frac{x-1}{x+1}\right)^n\frac{1}{\ln x} \,dx$
134787
How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$
83325
Integral of $\frac{\sqrt{x^2+1}}{x}$
16775
Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$
39073
Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
74955
Simplify result of $\int_0^{\infty} \frac{1}{1+x^n}dx$
63474
Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction?
48333
Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$
28771
Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $