Some Properties of Definite Integrals

Find $I$ where $I={\int }_0^{p+q\mathrm{\pi }}\left|\cos \mathit{x}\right|dx$ where $q\in N$ and $-\frac{\mathrm{\pi }}{2}<p<\frac{\mathrm{\pi }}{2}$.

Let $T>0$  be a fixed real number, suppose $f$ is a continuous function such that for all $x\in R, f\left(x+T\right)=f\left(x\right)$.

$I=\underset{0}{\overset{T}{\int }}f\left(x\right)dx$, then the value of $\underset{3}{\overset{3+3T}{\int }}f\left(2x\right)dx$ is

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

${\int }_0^{3a}\frac{f\left(x\right)}{f\left(x\right)+f(3a-x)}dx$

For $x\in R$, let $f\left(x\right)=\left|\sin x\right|$ and $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$. Let $p\left(x\right)=g\left(x\right)-\frac{2}{\pi }x$. Then

The value of the integral ${\int }_0^{\pi }\left(1-\left|\sin 8x\right|\right)dx$ is

The value of $I={\int }_0^3\left(\left[x\right]+\left[x+\frac{1}{3}\right]+\left[x+\frac{2}{3}\right]\right) dx,$ where $\left[·\right]$ denotes the greatest integer function, is equal to

If $A={\int }_0^1{x}^{50}{\left(2-x\right)}^{50}dx; B={\int }_0^1{x}^{50}{\left(1-x\right)}^{50} dx$ and $\frac{A}{B}=2^n,$ then the value of $n$ is

If $A={\int }_0^1{x}^{50}{\left(2-x\right)}^{50}dx; B={\int }_0^1{x}^{50}{\left(1-x\right)}^{50} dx$ and $\frac{A}{B}=2^n,$ find the value of $n$.

The value of the integral $\underset{-3\mathrm{\pi }}{\overset{3\mathrm{\pi }}{\int }}\left|{\mathrm{s}\mathrm{i}\mathrm{n}}^3\mathrm{}x\right|dx$ is equal to

${\int }_{-100\mathrm{\pi }}^{100\pi }\left({\sin }^{4}x+{\cos }^{4}x\right)dx$ is equal to :

$\underset{0}{\overset{100\pi }{\int }}\left(\sum _{r=1}^{10}\tan \hspace{0.17em}rx\right)dx$ is equal to

Evaluate the following:

${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\left(\sin \right|x|+\cos |x\left|\right)dx$

If ${\int }_{-2}^3\left|1-x^2\right|dx=\frac{k}{3}$, then what is the value of $k$.

Evaluate the integral: ${\int }_{-5}^5\left(x^3+2x\right)dx$.

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

${\int }_0^{2a}\frac{f\left(x\right)}{f\left(x\right)+f\left(2a-x\right)}dx$ is equal to

If ${\int }_{-1}^{4}f\left(x\right)dx=4$ and ${\int }_2^4\left[3-f\left(x\right)\right]dx=7,$ then the value of ${\int }_{-1}^{2}f\left(x\right)dx$ is