Properties of Definite Integrals

Evaluate :  $\underset{0}{\overset{\pi }{\int }}\frac{x\sin x}{1+{\cos }^{2}x}dx$

The value of integral ${\int }_0^{\mathrm{\pi }}xf\left(\sin x\right)dx=$

$I_1={\int }_0^{\pi }\frac{x\sin x}{1+{\cos }^{2}x}dx, I_2={\int }_0^{\pi }\frac{x^3\sin x}{\left({\pi }^2-3\pi x+3x^2\right)\left(1+{\cos }^{2}x\right)}dx,$ then

Let $I_n={\int }_0^{\frac{1}{2}}\frac{1}{\sqrt{1-x^n}}dx$ where $n>2$, then

Given $f$ is an odd function defined everywhere, periodic with period $2$ and integrable on every interval. Let $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$, then

Let $a$ be a real number in the interval $\left[0, 314\right]$ such that ${\int }_{-\pi +a}^{3\pi +a}\left|x-a-\pi \right|\sin \left(\frac{x}{2}\right)dx=-16$, then determine the number of such values of $a$.

Let $I_1={\int }_0^{\frac{\pi }{4}}{\left(1+\tan x\right)}^{2}dx, I_2={\int }_0^1\frac{dx}{{\left(1+x\right)}^2\left(1+x^2\right)},$ then find the value of $\frac{I_1}{I_2}$

Let $f\left(x\right)$ be a function satisfying $f\left(x\right)=f\left(\frac{100}{x}\right)\forall x>0.$ If ${\int }_1^{10}\frac{f\left(x\right)}{x}dx=5$, then find the value of ${\int }_1^{100}\frac{f\left(x\right)}{x}dx$

Let $f\left(x\right)$ and $g\left(x\right)$ be two functions satisfying $f\left(x^2\right)+g(4-x)=4x^3$ and $g\left(4-x\right)+g\left(x\right)=0,$ then the value of ${\int }_{-4}^{4}f\left(x^2\right)dx$ is

If the integral ${\int }_0^{10}\frac{\left[\sin 2\pi x\right]}{{\mathrm{e}}^{x-\left[x\right]}}dx=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma ,$ where $\alpha , \beta , \gamma$ are integers and $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ then the value of $\alpha +\beta +\gamma$ is equal to:

${\int }_{-\pi }^{\pi }\frac{2x\left(1+\sin x\right)}{1+{\cos }^{2}x}dx$ is

The value of ${\int }_{\sqrt{{\log }_{e}2}}^{\sqrt{{\log }_{e}3}}\frac{x \sin x^2}{\sin x^2+\sin \left({\log }_{e}6 - x^2\right)}dx$ is

Let $f\left(x\right)={\int }_0^x\frac{e^t}{t}dt\left(x>0\right)$, then $e^{-a}\left[f\right(x+a)-f(1+a\left)\right]=$

A function $f$ is defined by $f\left(x\right)={\int }_0^{\mathrm{\pi }}\cos t\cos (x-t)dt$, $0\le x\le 2\mathrm{\pi }$. Then which of the following hold(s) good?

If $A(x+y)=A\left(x\right)A\left(y\right) \forall x,y$ and $A\left(0\right)\ne 0$ and $B\left(x\right)=\frac{A\left(x\right)}{1+{\left(A\left(x\right)\right)}^2}$, then

$f:[0,5]\to R, y=f\left(x\right)$ such that $f^{\text{'}\text{'}}\left(x\right)=f^{\text{'}\text{'}}(5-x) \forall x\in [0,5], f^{\text{'}}\left(0\right)=1$ and $f^{\text{'}}\left(5\right)=7$, then the value of ${\int }_1^4{f}^{\text{'}}\left(x\right)dx$ is

A function $f$ is defined by $f\left(x\right)={\int }_0^{\pi }\mathrm{c}\mathrm{o}\mathrm{s}t \mathrm{c}\mathrm{o}\mathrm{s}\left(x-t\right)dt, 0\le x\le 2\mathrm{\pi }$, then which of the following hold(s) good

Let $f\left(x\right)$ be an odd continuous function which is periodic with period $2$. If $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$, then

The value of ${\int }_0^{\left[x\right]}(x-[x\left]\right)dx$ (where $[.]$ denotes the greatest integer function), is

The value of $\underset{0}{\overset{1}{\int }}\frac{8\log \left(1+x\right)}{1+x^2}dx$ is