Definite Integration

The value of $\underset{\mathrm{a}}{\overset{b}{\int }}f\left(x\right)dx=\left(b-a\right)\underset{n\to \infty }{\lim }\frac{1}{n}[f\left(a\right)+f\left(a+h\right)+\mathrm{...}+f\left(a+\left(n-1\right)h\right)],$ where $h=\frac{b-a}{n}$, $f\left(x\right)=x^2+x+2; a=0, b=2$ as limits of sum would be

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

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 

The value of $\underset{n\to \infty }{\lim }\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{6n}\right)$ is

$\underset{n\to \infty }{\lim }\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{6n}\right)$ is equal to

If $L=\underset{n\to \infty }{\lim }\left(\frac{1^5+2^5+3^5+....+n^5}{n^6}\right)+\underset{n\to \infty }{\lim }\left(\frac{1^4+2^4+3^4+....+n^4}{n^5}\right)-\underset{n\to \infty }{\lim }\left(\frac{1^3+2^3+....+n^3}{n^4}\right)$, then the value of $L$ is

Evaluate:${\int }_{\sqrt{3}}^{\sqrt{8}}x\sqrt{1+x^2}dx=?$

Evaluate:${\int }_{\sqrt{3}}^{\sqrt{8}}x\sqrt{1+x^2}dx=?$

If${\int }_0^{a}4x^{3}dx=16$, then find $a$.

Evaluate ${\int }_2^3\frac{x}{x^2-1}dx$.

Evaluate ${\int }_4^9\frac{1}{\sqrt{x}}dx$

Find the value of ${\int }_1^2\left(1+{\sin }^{2}x\right)dx+{\int }_1^2\left({\cos }^{2}x-1\right)dx$.

The integral ${\int }_0^{\frac{\mathrm{\pi }}{2}}{\sec }^{2}xdx-{\int }_0^{\frac{\mathrm{\pi }}{2}}{\tan }^{2}xdx$ is of the form $\frac{\mathrm{\pi }}{a}$, then $a=$

${\int }_0^1{\sin }^{-1}xdx+{\int }_0^1{\cos }^{-1}xdx=?$
Express the answer how many times of $\mathrm{\pi }$

Evaluate ${\int }_0^{\frac{\pi }{2}}{\cos }^{2}x \mathrm{d}x$.

Evaluate the following

${\int }_0^{\pi /2}\frac{1}{5+4\sin x}dx$

Evaluate the following

${\int }_0^{\pi /2}\frac{dx}{4\cos x+9\sin x}$

Evaluate the following integral as a limit of sum :

${\int }_1^3{x}^{3}dx$

Evaluate the following integral as a limit of sum :

${\int }_0^3(e^x+6x^2)dx$