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Integral $\int_{0}^{\infty} \frac{\left(\frac{\pi x}{2}-\log (x)\right)^3}{\left(x^2+1\right)^2 \left(\log^2(x)+\frac{\pi ^2}{4}\right)} = \pi$ proof

Define $I_n=\int_0^1\frac{x^n}{\sqrt{x^2+1}}dx$ for every $n\in\mathbb{N}$. Prove that $\lim_{n\to\infty}nI_n=\frac{1}{\sqrt 2}$.

Proof $\int_0^\infty \frac{\exp(-\sqrt{x^2+y^2})}{x^2+y^2}dx = \frac{\pi}{2}\left(\frac{1}{y} - K_0(y)L_{-1}(y) - K_1(y)L_{0}(y)\right)$

Evaluate $\int_0^\infty \frac{dx}{(x+\sqrt{1+x^2})^2}$

Evaluation of $\int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$

How do I evaluate this definite integral?

Calculating a limit using dominated convergence theorem

Double Integral $\int\limits_0^a\int\limits_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac32}$

Generalized Sophomore's dream. Question about originality

How to prove this inequality about the arc-length of convex functions?

How to evaluate $\int_0^\pi \cos(x) \cos(2x) \cos(3x) \cos(4x)\, dx$

Is $\pi/e$ a period?

Seeking Methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $

When calculating integrals, why replacing factorials with $\Gamma$ so often works?

How to evaluate $\int_0^\infty \frac{\cos(ax)}{1+x^N}dx$ for $a,N\in\mathbb{R}$ and $N> 1$

Integral of a floor function.

Substitution for definite integrals

Evaluating $\int _0^1\frac{\ln ^2\left(x\right)\ln \left(1-x\right)}{1+x^2}\:dx$

New posts in definite-integrals

Integral $\int_{0}^{\infty} \frac{\left(\frac{\pi x}{2}-\log (x)\right)^3}{\left(x^2+1\right)^2 \left(\log^2(x)+\frac{\pi ^2}{4}\right)} = \pi$ proof

Evaluate the limit of $(n+1)\int_0^1x^n\ln(1+x)\,dx$ when $n\to\infty$

Computing $ \int_0^{\infty} \frac{ \sqrt [3] {x+1}-\sqrt [3] {x}}{\sqrt{x}} \mathrm dx$

Define $I_n=\int_0^1\frac{x^n}{\sqrt{x^2+1}}dx$ for every $n\in\mathbb{N}$. Prove that $\lim_{n\to\infty}nI_n=\frac{1}{\sqrt 2}$.

Proof $\int_0^\infty \frac{\exp(-\sqrt{x^2+y^2})}{x^2+y^2}dx = \frac{\pi}{2}\left(\frac{1}{y} - K_0(y)L_{-1}(y) - K_1(y)L_{0}(y)\right)$

Evaluate $\int_0^\infty \frac{dx}{(x+\sqrt{1+x^2})^2}$

Evaluation of $\int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$

How do I evaluate this definite integral?

Calculating a limit using dominated convergence theorem

Double Integral $\int\limits_0^a\int\limits_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac32}$

Generalized Sophomore's dream. Question about originality

How to prove this inequality about the arc-length of convex functions?

How to evaluate $\int_0^\pi \cos(x) \cos(2x) \cos(3x) \cos(4x)\, dx$

Is $\pi/e$ a period?

Seeking Methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $

When calculating integrals, why replacing factorials with $\Gamma$ so often works?

How to evaluate $\int_0^\infty \frac{\cos(ax)}{1+x^N}dx$ for $a,N\in\mathbb{R}$ and $N> 1$

Integral of a floor function.

Substitution for definite integrals

Evaluating $\int _0^1\frac{\ln ^2\left(x\right)\ln \left(1-x\right)}{1+x^2}\:dx$