Newbetuts

How to show the following integral $\int_0^\frac{\pi}{2} \cot^{-1}{\sqrt{1+\csc{\theta}}}\,\mathrm d\theta =\frac{\pi^2}{12}$

Smart Integration Tricks [closed]

Is it possible to compute $\int_{-\infty}^\infty {x \sin x\over x^4+1}dx$ without using complex analysis?

Integral $\int_0^\infty\sin{(x^4)} dx$

Is there a closed form expression for $\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )} \mathrm{d}x\,\mathrm{d}y$?

Ways to prove that $\int_0^\infty \frac{x^2 \ln(x)}{(1+x^2)^3} {\rm d}x = 0$.

Regarding integrals of type $f(x)=\int_{a}^{b}g{(x+f(x))}dx$

Evaluate $ \int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx $

Show that $\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx = \frac{8 \pi ^3}{81 \sqrt{3}}$

How can we show that $\int_{0}^{2\pi}{x\over \phi-\cos^2(x)}\mathrm dx=2\pi^2?$

How to solve $\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx$?

Easiest way to find $\Re\int_{0}^{\pi/2} e^{e^{i\theta}} d\theta$

Calculate $\int _0^\infty \frac{\ln x}{(x^2+1)^2}dx$

Expression for $\int_0^1 x^n(1-x)^{n}/(1+x^2) \ dx$

Maximum and minimum value of $\int_0^1 f(x)dx $ given $|f'(x)|<2$

How to calculate $\int_0^{\frac{\pi}{2}}{x^2\cot x\ln\cos x\, \text{d}x}$.

New posts in definite-integrals

Computing $\int_0^\infty\mathrm{d} x\frac{x}{e^x+1}$ with contour integration

Prove $\int_0^1\frac{\ln2-\ln\left(1+x^2\right)}{1-x}\,dx=\frac{5\pi^2}{48}-\frac{\ln^22}{4}$

Is there a standard way to write a second definite integral?

Evaluating $\lim_{n\rightarrow\infty} \int_{0}^{\pi} \frac {\sin x}{1+ \cos^2 (nx)} dx$

How to show the following integral $\int_0^\frac{\pi}{2} \cot^{-1}{\sqrt{1+\csc{\theta}}}\,\mathrm d\theta =\frac{\pi^2}{12}$

Smart Integration Tricks [closed]

Is it possible to compute $\int_{-\infty}^\infty {x \sin x\over x^4+1}dx$ without using complex analysis?

Integral $\int_0^\infty\sin{(x^4)} dx$

Is there a closed form expression for $\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )} \mathrm{d}x\,\mathrm{d}y$?

Ways to prove that $\int_0^\infty \frac{x^2 \ln(x)}{(1+x^2)^3} {\rm d}x = 0$.

Regarding integrals of type $f(x)=\int_{a}^{b}g{(x+f(x))}dx$

Evaluate $ \int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx $

Show that $\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx = \frac{8 \pi ^3}{81 \sqrt{3}}$

How can we show that $\int_{0}^{2\pi}{x\over \phi-\cos^2(x)}\mathrm dx=2\pi^2?$

How to solve $\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx$?

Easiest way to find $\Re\int_{0}^{\pi/2} e^{e^{i\theta}} d\theta$

Calculate $\int _0^\infty \frac{\ln x}{(x^2+1)^2}dx$

Expression for $\int_0^1 x^n(1-x)^{n}/(1+x^2) \ dx$

Maximum and minimum value of $\int_0^1 f(x)dx $ given $|f'(x)|<2$

How to calculate $\int_0^{\frac{\pi}{2}}{x^2\cot x\ln\cos x\, \text{d}x}$.