Properties of Definite Integrals

If $f\left(x\right)$ is a function such that $f\left(x\right)$. $f\left(-x\right)=9$ $\forall x\in$ $\left[-51,51\right]$, then ${\int }_{-51}^{51}\frac{dx}{3+f\left(x\right)}$ has the value equal to

$f\left(x\right)=\sin \mathit{x}+{\int }_0^x{f}^{\text{'}}\left(t\right)\left(2\sin \mathit{t}-{\sin }^{2}t\right)dt$, then $f\left(x\right)$ is

Find ${\int }_0^{\infty }\text{f}\left(\frac{\text{a}}{\text{x}}+\frac{\text{x}}{\text{a}}\right)·\frac{\ell \text{nx}}{\text{x}}\text{dx}$

The value of  $\underset{x\to \infty }{\lim }\left(x^3{\int }_{-\frac{1}{x}}^{\frac{1}{x}}\frac{\ln \left(1+t^2\right)}{1+e^t}dt\right)$ is

The value of ${\int }_0^{\infty }\frac{dx}{x^2+2x \cos \theta +1}$ is 

For $\text{f}\left(\text{x}\right)={\text{x}}^4+\left|\text{x}\right|$, let ${\text{I}}_1={\int }_0^{\pi }\text{f}\left(\text{cos x}\right)\text{ dx}$ and ${\text{I}}_2={\int }_0^{\pi /2}\text{f}\left(\text{sin x}\right)\text{ dx }$then $\frac{{\text{I}}_1}{{\text{I}}_2}$ has the value of equal to

${\int }_0^{\infty }\frac{dx}{\left(1+x^a\right)\left(1+x^2\right)}; \left(a>0\right)$

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 ${\text{I}}_{\text{n}}={\int }_{-\text{n}}^{\text{n}}\left(\{\text{x}+1\}\text{.}\{{\text{x}}^2+2\}+\{{\text{x}}^2+2\}\{{\text{x}}^3+4\}\right)\text{dx,}$ given $(\text{n}\in N)$ where $\{.\}$ denotes fractional part of $\text{x},$ then ${\text{I}}_1=$

$\text{I}={\int }_0^{\pi }\frac{\left(\text{ax}+\text{b}\right)\text{sec}\text{x}\text{tan}\text{x}}{4+{\text{tan}}^2\text{x}}\text{dx}\left(\text{a, b}>0\right)$, then I is equal to

${\int }_0^{\frac{\pi }{4}}\log \left(1+\tan x\right)dx=$

If $\underset{0}{\overset{a}{\int }}f\left(x\right)dx=\underset{0}{\overset{a}{\int }}f(a-x)dx$, then the value of $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{dx}{1+\tan x}$ is

The value of  ${\displaystyle {\int }_2^3\frac{\sqrt{x}}{\sqrt{\mathrm{}\mathrm{}\mathrm{}5-x}+\sqrt{x}}dx}$  is:

The value of  ${\displaystyle \underset{\pi /4}{\overset{\mathrm{}3\pi /4}{\int }}\frac{\varphi }{1+\sin \varphi }d\varphi }$  is:

The value of  ${\int }_{-2}^2|1-x^2|dx$  is:

$f(x)=\left|\begin{array}{ccc}\sec x& \cos x& {\sec }^{2}x+\cot x\mathrm{cosec }\mathrm{x }\\ {\mathrm{ cos}}^{2}x& {\cos }^{2}x& {\mathrm{cosec}}^{2}x\\ 1& {\cos }^{2}x& {\cos }^{2}x\end{array}\right|.$

Then  ${\displaystyle \underset{0}{\overset{\pi /2}{\int }}f(x)dx\mathrm{equals}:}$