${\int }_0^2\left[{\mathrm{x}}^2\right]\mathrm{dx}$ is equal to

• $2 - \sqrt{2}$

• $2 + \sqrt{2}$

• $\sqrt{2} - 1$

• $- \sqrt{2} - \sqrt{3} + 5$

In = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\tan }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$, then $\underset{\mathrm{n}\to \infty }{\lim }\mathrm{n}\left[{\mathrm{I}}_{\mathrm{n}} + {\mathrm{I}}_{\mathrm{n} + 2}\right]$ equals

• 1/ 2

• 2 sq units

• 3 sq units

• 4 sq uits

${\int }_0^{10\mathrm{\pi }}\left|\sin \left(\mathrm{x}\right)\right|\mathrm{dx}$ is equal to

• 20

• 8

• 10

• 18

If I = ${\int }_{{\mathrm{x}}_0}^{{\mathrm{x}}_0 + \mathrm{nh}}\mathrm{ydx}$, then by Trapezoidal rule I is equal to

• $\mathrm{h}\left[\left({\mathrm{y}}_0 + {\mathrm{y}}_{\mathrm{n}}\right) + 2\left({\mathrm{y}}_1 + {\mathrm{y}}_2 + ... + {\mathrm{y}}_{\mathrm{n} - 1}\right)\right]$

• $\mathrm{h}\left[\frac{1}{2}\left({\mathrm{y}}_0 + {\mathrm{y}}_{\mathrm{n}}\right) + 2\left({\mathrm{y}}_1 + {\mathrm{y}}_2 + ... + {\mathrm{y}}_{\mathrm{n} - 1}\right)\right]$

• $\frac{\mathrm{h}}{2}\left[\left({\mathrm{y}}_0 + {\mathrm{y}}_{\mathrm{n}}\right) + 2\left({\mathrm{y}}_1 + {\mathrm{y}}_2 + ... + {\mathrm{y}}_{\mathrm{n} - 1}\right)\right]$

• $\mathrm{h}\left[\left({\mathrm{y}}_0 + {\mathrm{y}}_{\mathrm{n}}\right) + 2\left({\mathrm{y}}_1 + {\mathrm{y}}_2 + ... + {\mathrm{y}}_{\mathrm{n} - 1}\right)\right]$

$\int \frac{\mathrm{dx}}{1 - {\mathrm{x}}^2}$ is equal to

• ${\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{c}$

• ${\sin }^{-1}\left(\mathrm{x}\right) + \mathrm{c}$

• $\frac{1}{2}\log \left|\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right| + \mathrm{c}$

$\int \frac{\cos \left(2\mathrm{x}\right) - 1}{\cos \left(2\mathrm{x}\right) + 1}\mathrm{dx}$ is equal to

• $\tan \left(\mathrm{x}\right) - \mathrm{x} + \mathrm{c}$

• $\mathrm{x} + \tan \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{x} - \tan \left(\mathrm{x}\right) + \mathrm{c}$

• $- \mathrm{x} - \cot \left(\mathrm{x}\right) + \mathrm{c}$

547.

Suppose f is such that f( - x) = - f(x), for every real x and ${\int }_0^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = 5, \mathrm{then} {\int }_{- 1}^0\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$ is equal to

• 10

• 5

• 0

• - 5

If f(y) = e^y, g(y) = y, y > 0 and F(t) = ${\int }_0^{\mathrm{t}}\mathrm{f}\left(\mathrm{t} - \mathrm{y}\right) . \mathrm{g}\left(\mathrm{y}\right)\mathrm{dy}$, then 

• F(t) = 1 - e^{- t}(1 + t)

• F(t) = e^t - (1 + t)

• F(t) = te^t

• F(t) = te^{- t}

Let f(x) be a function satisfying f'(x) = f(x) with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = x². Then, the value of the integeral ${\int }_0^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}$ is

• $\mathrm{e} - \frac{{\mathrm{e}}^2}{2} - \frac{5}{2}$

• $\mathrm{e} + \frac{{\mathrm{e}}^2}{2} - \frac{3}{2}$

• $\mathrm{e} - \frac{{\mathrm{e}}^2}{2} - \frac{3}{2}$

• $\mathrm{e} + \frac{{\mathrm{e}}^2}{2} + \frac{5}{2}$

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

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{- \mathrm{\pi }}{4}$