Properties of Definite Integrals

Direction

Let $I_n={\int }_{-\pi }^{\pi }\frac{{\sin }^2 x}{(1+{\pi }^x)}dx; n=0, 1, 2...$

The value of $\sum _{m=1}^{10} I_{2m}$ is equal to

Direction

Let $I_n={\int }_{-\pi }^{\pi }\frac{\sin x}{(1+{\pi }^x)sinx}dx; n=0, 1, 2...$

The value of $\sum _{m=1}^{10} I_{2m+1}$ is equal to

Direction:

Let $f\left(x\right)=\left\{\begin{array}{l}1-\left|x\right|, \left|x\right|\le 1\\ 0, \left|x\right|>1\end{array}\right.$ and $g\left(x\right)=f(x+1)+f(x-1)$ for all $x\in R$.

The value of ${\int }_{-3}^3 g\left(x\right)dx$ is

The value of ${\int }_{-2}^2\left|\left[x\right]\right|dx$ is equal to

Let $f:R\to R$ be a continuous function given by $f(x+y)=f\left(x\right)+f\left(y\right)$ for all $x, y\in R$. If ${\int }_0^{2}f\left(x\right)dx=\alpha$, then ${\int }_{-2}^{2}f\left(x\right)dx$ is equal to  

Let $f\left(x\right)=\frac{e^x+1}{e^x-1}$ and ${\int }_0^1{x}^3·\frac{e^x+1}{e^x-1}dx=\alpha$. Then ${\int }_{-1}^1{t}^{3}f\left(t\right)dt$ is equal to

Let $f\left(x\right)$ be a continuous function such that  ${\int }_n^{n+1}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=n^3, n\in \mathrm{Z}$. Then, ${\int }_{-3}^3 \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is 

The value of ${\int }_{-2}^2\frac{{\sin }^{2}x}{\left[{\displaystyle \frac{x}{\mathrm{\pi }}}\right]+{\displaystyle \frac{1}{2}}}dx$, where $[.]$ denotes greatest integer function, is

The equation ${\int }_{-\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{4}}\left\{a\left|\sin x\right|+\frac{b\sin x}{1+{\cos }^{2}x}+c\right\}dx=0$, where $a, b$ and $c$ are constants, gives a relation between

The value of ${\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\frac{1}{e^{\sin x}+1}dx$ is equal to

The integral ${\int }_{-\frac{1}{2}}^{\frac{1}{2}}\left(\left[x\right]+{\log }_e\left(\frac{1+x}{1-x}\right)\right)dx$ (where, [.] denotes greatest integer function) is equal to

The value of ${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\frac{{\cos }^{2}x}{1+e^x}dx$ is:

The value of ${\int }_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}}\frac{{\cos }^{-1}\left({\displaystyle \frac{2x}{1+x^2}}\right)+{\tan }^{-1}\left(\frac{2x}{1-x^2}\right)}{1+e^x}dx$ is equal to

If $f\left(x\right)$ is an odd function, then the value of ${\int }_{-\mathrm{a}}^{\mathrm{a}}\frac{f\left(\sin x\right)}{f\left(\cos x\right)+f\left({\sin }^{2}x\right)}dx$ ie equal to

The value of ${\int }_{-1}^1\left(\frac{{\mathrm{x}}^2+\mathrm{sinx}}{1+{\mathrm{x}}^2}\right)dx$ is equal to

The value of ${\int }_{-1}^1\left(x\left|x\right|\right)dx$ is equal to

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

If ${\int }_0^x\left[x\right]dx={\int }_0^{\left[x\right]}xdx, x\notin$integer (where $\left[.\right]$ and $\left\{.\right\}$ denotes the greatest integer and fraction parts respectively), then the value of $4\left\{x\right\}$ is equal to

If $f\left(n\right)={\int }_0^x\left[\cos t\right]dt$, where $x\in \left(2n\mathrm{\pi },2n\mathrm{\pi }+\frac{\mathrm{\pi }}{2}\right);n\in N$ and $\left[.\right]$ denotes the greatest integer function, then the value of $\left|f\left(\frac{1}{\mathrm{\pi }}\right)\right|$ is 

 If $f\left(n\right)=\frac{{\int }_0^{\mathrm{n}}\left[x\right]dx}{{\int }_0^{\mathrm{n}}\left\{x\right\}dx}$ (where  $[.]$ and $\{.\}$ denotes greatest integer and fractional part of$x$ and $n\in N$). Then, the value of $f\left(4\right)$ is