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Closed-form of $\int\limits_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$

Discussing the Integral of $\exp(-x^n)$

Is $\frac{1}{\pi}\int_{0}^{\infty} \Gamma(\sigma +xi)^2\,\Gamma(\sigma-xi)^2 \,dx = \frac{\Gamma(2\,\sigma)^4}{\Gamma(4\,\sigma)}$?

Evaluate $\int_0^1{\frac{y}{\sqrt{y(1-y)}}dy}$

Integral $\int_0^1 \left(\arctan x \right)^2\,dx$

Is there any trick to finding $\int_0^\pi [4 \cos (6x) + \cos(7x)] \cos(nx) dx$?

Compute $\int_0^{\pi} \frac{\cos(nx)\cos(x) - \cos(nt)\cos(t)}{\cos(x) -\cos(t)}dt$

How to prove $\int_0^{2\pi} \ln(1+a^2+2a\cos x)\, dx=0$? [duplicate]

Compute $\int_0^\infty\frac{x^a-x^b}{(1+x^a)(1+x^b)}\,dx$

Evaluate $\int_0^\pi \frac{\sin\frac{21x}{2}}{\sin \frac x2} dx$ (from MIT Integration Bee)

What's the value of $\int_0^1\frac{1}{2y} \ln(y) \ln^2(1-y) \, dy$?

How to show $\int_0^\infty\frac1{(1+x^2)(1+x^p)}$ doesn't depend on $p$?

Integral $\int_0^{\pi/2} \frac{\sin^3 x\log \sin x}{\sqrt{1+\sin^2 x}}dx=\frac{\ln 2 -1}{4}$

Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

Integral $\int_0^1 \frac{\ln(1-x)\ln(1+x^2)}{x}dx$

Parseval's Identity (Integral)

New posts in definite-integrals

Integral of $\int_0^{\pi/2} \frac {\sqrt{\sin x}}{\sqrt {\sin x} + \sqrt {\cos x}}$dx [duplicate]

An equivalent for $\sum_{n=0}^{\infty} e^{-x\sqrt{n}}$ as $x$ tends to $0^+$

Is there a closed form for $\int_0^1\frac{\ln^4(1-x)\operatorname{Li}_4(x)}{x}dx\ ?$

Finding the following definite and indefinite integrals

Closed-form of $\int\limits_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$

Discussing the Integral of $\exp(-x^n)$

Is $\frac{1}{\pi}\int_{0}^{\infty} \Gamma(\sigma +xi)^2\,\Gamma(\sigma-xi)^2 \,dx = \frac{\Gamma(2\,\sigma)^4}{\Gamma(4\,\sigma)}$?

Evaluate $\int_0^1{\frac{y}{\sqrt{y(1-y)}}dy}$

Integral $\int_0^1 \left(\arctan x \right)^2\,dx$

Is there any trick to finding $\int_0^\pi [4 \cos (6x) + \cos(7x)] \cos(nx) dx$?

Compute $\int_0^{\pi} \frac{\cos(nx)\cos(x) - \cos(nt)\cos(t)}{\cos(x) -\cos(t)}dt$

How to prove $\int_0^{2\pi} \ln(1+a^2+2a\cos x)\, dx=0$? [duplicate]

Compute $\int_0^\infty\frac{x^a-x^b}{(1+x^a)(1+x^b)}\,dx$

Evaluate $\int_0^\pi \frac{\sin\frac{21x}{2}}{\sin \frac x2} dx$ (from MIT Integration Bee)

What's the value of $\int_0^1\frac{1}{2y} \ln(y) \ln^2(1-y) \, dy$?

How to show $\int_0^\infty\frac1{(1+x^2)(1+x^p)}$ doesn't depend on $p$?

Integral $\int_0^{\pi/2} \frac{\sin^3 x\log \sin x}{\sqrt{1+\sin^2 x}}dx=\frac{\ln 2 -1}{4}$

Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

Integral $\int_0^1 \frac{\ln(1-x)\ln(1+x^2)}{x}dx$

Parseval's Identity (Integral)