Definite Integration Questions with Solutions

Evaluate ${\int }_0^{\pi /4}\frac{dx}{{cos}^{4}x}$

If $f:[0,\infty )\to R$ is defined by $f\left(x\right)=\frac{x^2-1}{x^2+1}$, find the value of ${\int }_{-1}^1{f}^{-1}\left(y\right)dy$.

The value of the integral ${\int }_0^1\frac{e^{5{log}_{e}x}-e^{4{log}_{e}x}}{e^{{log}_e{x}^3}-e^{{log}_e{x}^2}}dx$ is

The value of the integral ${\int }_{-2}^0\frac{dx}{\sqrt{12-x^2-4x}}$ is

The value of ${\int }_{-\pi /3}^{\pi /3}\frac{xsinx}{{cos}^{2}x}dx$ is

If $I_n={\int }_0^{\pi /4}{tan}^n\theta d\theta$, then $I_8+I_6$ equals

The value of the integral ${\int }_0^{\pi /2}\frac{\sqrt{sinx}}{\sqrt{sinx}+\sqrt{cosx}}dx$ is

The value of ${lim}_{n\to \infty }\frac{\pi }{n}\left[sin\frac{\pi }{n}+sin\frac{2\pi }{n}+\cdots +sin\frac{\left(n-1\right)\pi }{n}\right]$ is

If $I_1={\int }_0^1{2}^{x^2}dx$, $I_2={\int }_0^1{2}^{x^3}dx$, $I_3={\int }_1^2{2}^{x^2}dx$ and $I_4={\int }_1^2{2}^{x^3}dx$, then

The value of the integral ${\int }_0^{\pi /2}logtanxdx$ is

For $0\le x\le \frac{\pi }{2}$, the value of ${\int }_0^{{sin}^{2}x}{sin}^{-1}\sqrt{t}dt+{\int }_0^{{cos}^{2}x}{cos}^{-1}\sqrt{t}dt$ is

If $a$ is a positive integer, the number of values of $a$ satisfying ${\int }_0^{\pi /2}\left[a^2\left(\frac{cos3x}{4}+\frac{3}{4}cosx\right)+asinx-20cosx\right]dx\le -\frac{a^2}{3}$ is

If $f\left(x\right)={\int }_0^{x}tsintdt$, then $f\prime \left(x\right)$ is

${\int }_0^{1/2}\frac{dx}{\sqrt{x-x^2}}$ is equal to

The value of the integral ${\int }_3^6\frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}}dx$ is

The value of the integral ${\int }_0^{\pi /4}\frac{sinx+cosx}{3+sin2x}dx$ is

If $I_1={\int }_0^{1}2x^{2}dx$, $I_2={\int }_0^{1}2x^{3}dx$, $I_3={\int }_1^2{x}^{2}dx$ and $I_4={\int }_1^{2}2x^{3}dx$, then

The value of ${\int }_0^{\pi }\frac{xsinx}{1+{cos}^{2}x}dx$ is:

The value of ${\int }_0^{\pi /2}\frac{dx}{1+{tan}^{3}x}$ is

If $f:R\to R$ and $g:R\to R$ are continuous functions, then the value of ${\int }_{-\pi /2}^{\pi /2}\left[f\left(x\right)+f\left(-x\right)\right]\left[g\left(x\right)-g\left(-x\right)\right]dx$ is