Definite Integration

${\int }_{-\frac{\mathrm{\pi }}{3}}^{\frac{\mathrm{\pi }}{3}}{\sin }^2 \mathrm{xdx}=?$

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 $A\left(x\right)={\int }_0^x{\theta }^{2}d\theta$ , then the value of  $A\left(3\right)$ is

The value of ${\int }_{a-c}^{b-c}f(x+c)dx$ is

The value of ${\int }_2^5\frac{\sqrt{x}}{\sqrt{x}+\sqrt{7-x}}dx$ is

The value of ${\int }_0^{\frac{\mathrm{\pi }}{4}}\sqrt{1+\sin 2x}dx$ is

If $a=(3t^2+2t+1 ) m s^{-2}$ is the expression according to which the acceleration of a particle varies. Then, the change in velocity after $3$ $\mathrm{s}$ of its start is

If  $a=\left(3t^2+2t+1\right){\mathrm{ms}}^{-2}$ is the expression according to which the acceleration of a particle varies. 

The expression for instantaneous velocity at any time $t$ will be, (if the particle was initially at rest)

The value of $\underset{0}{\overset{1}{\int }}{\cos }^{-1}\hspace{0.17em}x\hspace{0.17em}dx\hspace{0.17em}\mathrm{+}\underset{0}{\overset{1}{\int }}{\cos }^{-1}\left(\sqrt{1-x^2}\right)\hspace{0.17em}dx$ is

The value of $I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{{\left(\mathrm{s}\mathrm{i}\mathrm{n}x+\mathrm{c}\mathrm{o}\mathrm{s}x\right)}^2}{\sqrt{1+\mathrm{s}\mathrm{i}\mathrm{n}2x}}dx$ is

The value of ${\int }_0^{\pi }\left|{\sin }^3\theta \right|d\theta$ is

$\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\left[\frac{1}{n+1}+\frac{1}{n+2}+\dots ..+\frac{1}{6n}\right]$ is equal to

Let, $I=\underset{a}{\overset{b}{\int }}\left(x^4-2x^2\right)dx$, then the ordered pair $\left(a,b\right)$ for which the value of $I$ is minimum $\left(b>0\right)$ is

The value of $\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{1}{n}\left\{\sqrt[n]{5}+\sqrt[n]{5^2}+...+\sqrt[n]{5^n}\right\}$ is equal to

$\underset{0}{\overset{1}{\int }}\frac{dx}{e^x+e^{-x}}$ is equal to

$\underset{-\frac{\pi }{4}}{\overset{\frac{\pi }{4}}{\int }}\frac{dx}{1+\cos 2x}$ is equal to

$\underset{-1}{\overset{0}{\int }}\frac{dx}{x^2 + x + 2}$ is equal to

${\int }_{-1}^{10}\mathrm{sgn}\left(x-\left[x\right]\right)dx$ $\left\{\text{where }\left[.\right]\text{ denotes the greatest integer function and }\mathrm{sgn} \text{denotes signum function}\right\}$ equal to

The value of the integral ${\int }_0^{0.9}\left[x-2\left[x\right]\right]dx,$ where $\left[\cdot \right]$ denotes the greatest integer function is