Basics 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

If $y=x^{\underset{1}{\overset{x}{\int }}\ln tdt}$, then the value of $\frac{dy}{dx}$ at $x=e$ is

If $\mathrm{\varphi }\left(x\right)=\cos x-\underset{0}{\overset{x}{\int }}\left(x-t\right)\mathrm{ \varphi }\left(t\right)dt.$ Then find the value of ${\varphi }^{\text{'}\text{'}}\left(x\right)+\varphi \left(x\right).$

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

$I={\int }_0^{\pi }\frac{x^{2 }\sin 2x\sin \left(\frac{\pi }{2}\cos x\right)}{2x-\pi } dx$

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

If ${\int }_{\frac{\pi }{4}}^{\frac{\pi }{3}}\frac{\left({\sin }^3\theta -{\cos }^3\theta -{\cos }^2\theta \right){\left(\sin \theta +\cos \theta +{\cos }^2\theta \right)}^{2007}}{{\left(\sin \theta \right)}^{2009}{\left(\cos \theta \right)}^{2009}}d\theta$$=\frac{{\left(a+\sqrt{b}\right)}^d-{\left(1+\sqrt{c}\right)}^d}{d}$, where $a, b, c$  and $d$ are all positive integers. Then the value of $(a+b+c+d)$ is

The value of : $\underset{0}{\overset{2}{\int }}\sqrt{4-x^2}dx$ would be:

The value of ${\int }_1^{e^{37}}\frac{\mathrm{\pi sin}\left(\mathrm{\pi }\ell nx\right)}{x}dx$ is

The value of  ${\int }_1^{e^{37}}\frac{\pi \sin \left(\pi \ln x\right)}{x}dx$  is:

Let $f$ be a non-negative function defined on the interval $\left[0,1\right].$ If  $\underset{0}{\overset{x}{\int }}\sqrt{1-{(f\text{'}(t\left)\right)}^2}dt=\underset{0}{\overset{x}{\int }}f\left(t\right)dt,0\le x\le 1,$  and  $f(0)=0,$  then:

The value of the integral ${\displaystyle \underset{0}{\overset{1}{\int }}\sqrt{\frac{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}1-x}{1+x}}dx}$ is

The value of the integral $\underset{0}{\overset{1}{\int }}\sqrt{\frac{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}1-x}{1+x}}dx$ is

The value of the integral $\underset{e^{-1}}{\overset{e^2}{\int }}\left|\frac{{\log }_{e}x}{x}\right|dx$ is

Let  $f\left(x\right)=x-\left[x\right],$  for every real number $x,$ where $\left[x\right]$ is the integral part of $x.$ Then  $\underset{-1}{\overset{1}{\int }}f\left(x\right)dx$ is:

If $f\left(x\right)=Asin\left(\frac{\pi x}{2}\right)+B, f\text{'}\left(\frac{1}{2}\right)=\sqrt{2} \mathrm{and}\underset{0}{\overset{1}{\int }}f\left(x\right)dx=\frac{2A}{\pi },$ then constants $A$ and $B$ are respectively 

$f\left(x\right)=A\sin \left(\frac{\pi x}{2}\right)+B, f^{\text{'}}\left(\frac{1}{2}\right)=\sqrt{2}$and${\int }_0^{1}f\left(x\right)dx=\frac{2A}{\mathrm{\pi }}$, then constants $A$ and $B$ are

Let $f$ be a positive function.

Let $I_1=\underset{1-k}{\overset{k}{\int }}xf\left[x\left(1-x\right)\right]dx, I_2=\underset{1-k}{\overset{k}{\int }}f\left[x\left(1-x\right)\right]dx,$ where $2k-1>0.$ Then $\frac{I_1}{I_2}$ is