Properties of Definite Integrals

The value of the integral  $\underset{0}{\overset{2a}{\int }}\frac{f\left(x\right)}{\left\{f\left(x\right)+f\left(2a-x\right)\right\}}dx$ is equal to

${\int }_{-3}^3{\cot }^{-1} \mathrm{x} d\mathrm{x}=$

If there exist a function $f\left(x\right)$ such that  $f\left(-x\right)=f\left(x\right)$ and $f\left(x+2\right)=f\left(x\right)$  and another function $g\left(x\right)$ is defined as $g\left(x\right)={\int }_0^{x}f\left(t\right) dt$and $g\left(1\right)=k$.

Then the value of $g\left(x+2\right)-g\left(x\right)$ is equal to

If the graph of the function $y=f\left(x\right)$ is represented by the line joining $\left(1,0\right)$ and $\left(0,1\right)$ in the plane, then $\underset{-1}{\overset{0}{\int }}f\left(x\right)dx$ is

$\underset{-2}{\overset{3}{\int }}\left|1-x^2\right|dx=$

${\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\left|\sin x\right|dx$ is

${\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^n\theta }{{\sin }^n\theta +{\cos }^n\theta }d\theta$ is equal to

Evaluate integral of $\frac{x^2}{\left(x^2+{\left(9-x\right)}^2\right)}$ with limits from $4$ to $5.$ The result is

The integral $\underset{0}{\overset{\pi }{\int }}\frac{x}{2\mathrm{c}\mathrm{o}\mathrm{s}ec x-\sin x}dx$ is equal to

$\underset{0}{\overset{\mathrm{\pi }}{\int }}{\sin }^{2015}x{\cos }^{2016}xdx$

The value of the integral ${\int }_{-2}^2\frac{{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}}dx$ (where $\left[x\right]$ denotes the greatest integer less than or equal to $x$ ) is

${\int }_{-2}^2\frac{{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}}dx$, where $\left[x\right]$ denotes greatest integer function less than or equal to $x$

$\underset{0}{\overset{2\pi }{\int }}{\sin }^{9}xdx$ is equal to

If $I=\underset{1}{\overset{4}{\int }}{\left(\left\{x\right\}\right)}^{\left[x\right]}dx$ then $\frac{24}{13}I$ is equal to

The value of the integral $\underset{-\pi }{\overset{\pi }{\int }}\frac{{\mathrm{c}\mathrm{o}\mathrm{s}}^{2}x}{1+a^x}dx,$ $a>0$ is

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\sin x-\cos x}{1+\sin x\cdot \cos x}dx$ is equal to

The value of $\underset{\frac{1}{n}}{\overset{\frac{\left(an-1\right)}{n}}{\int }}\frac{\sqrt{x}}{\sqrt{a-x}+ \sqrt{x}}dx$ is equal to

Evaluate: $\underset{-\pi }{\overset{199\pi }{\int }}\sqrt{\left(\frac{1-\cos 2x }{2}\right) }dx$

The value of $\underset{0}{\overset{1}{\int }}x {\left(1-x\right)}^{99} dx$ is 

Let $f$be a$$positive function. Let $I_1={\int }_{1-k}^{k}xf\left\{x\left(1-x\right)\right\}dx, I_2={\int }_{1-k}^{k}f\left\{x\left(1-x\right)\right\}dx$, where $2k-1>0.$Then$, \frac{I_1}{I_2}$ is