If ${\int }_{- 1}^4\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = 4 \mathrm{and} {\int }_2^4\left\{3 - \mathrm{f}\left(\mathrm{x}\right)\right\}d\mathrm{x} = 7$, then the value of

${\int }_{- 1}^2\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is

• - 2

• 3

• 4

• 5

If m, n be integers, then find the value of

${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}{\left(\cos \left(\mathrm{mx}\right) - \sin \left(\mathrm{nx}\right)\right)}^{2}d\mathrm{x}$

If I = ${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}\frac{{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}{{\mathrm{e}}^{\sin \left(\mathrm{x}\right)} +\hspace{0.17em}{\mathrm{e}}^{- \sin \left(\mathrm{x}\right)}}d\mathrm{x}$, then I equals

• $\frac{\mathrm{\pi }}{2}$

• $2\mathrm{\pi }$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

If h(x) = ${\int }_0^{\mathrm{x}}{\sin }^4\left(\mathrm{t}\right)d\mathrm{t}, \mathrm{then} \mathrm{h}\left(\mathrm{x} + \mathrm{\pi }\right)$ equals

• $\frac{\mathrm{h}\left(\mathrm{x}\right)}{\mathrm{h}\left(\mathrm{\pi }\right)}$

• $\mathrm{h}\left(\mathrm{x}\right)\mathrm{h}\left(\mathrm{\pi }\right)$

• $\mathrm{h}\left(\mathrm{x}\right) - \mathrm{h}\left(\mathrm{\pi }\right)$

• $\mathrm{h}\left(\mathrm{x}\right) + \mathrm{h}\left(\mathrm{\pi }\right)$

The value of the integral ${\int }_0^2\left|{\mathrm{x}}^2 - 1\right|d\mathrm{x}$ is

• 0

• 2

• - $\frac{1}{3}$

• - 2

The value of ${\int }_0^{\mathrm{\pi }}\left|\cos \left(\mathrm{x}\right)\right|d\mathrm{x}$ is

• $2\mathrm{\pi }$

• 2

• $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\pi }$}\right.$

• $\mathrm{\pi }$

The value of ${\int }_{- 3}^3\left({\mathrm{ax}}^5 + {\mathrm{bx}}^3 + \mathrm{cx} + \mathrm{k}\right)d\mathrm{x}$ where a, b, c, k are constant, depends only on

• a and k

• a and b

• a, b and c

• k

388.

The value of the integral ${\int }_{- \mathrm{a}}^{\mathrm{a}}\frac{{\mathrm{xe}}^{{\mathrm{x}}^2}}{1 + {\mathrm{x}}^2}d\mathrm{x}$ is

• ${\mathrm{e}}^{{\mathrm{a}}^2}$

• 0

• ${\mathrm{e}}^{- {\mathrm{a}}^2}$

• a

Evaluate $\int \frac{{\mathrm{x}}^2}{\mathrm{x}\left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)}\mathrm{dx}$

If $\int \frac{\sin \left(\mathrm{x}\right)}{\sin \left(\mathrm{x} - \mathrm{a}\right)}\mathrm{dx} = \mathrm{Ax} + \mathrm{Blog}\left(\sin \left(\mathrm{x} - \mathrm{\alpha }\right)\right) + {\mathrm{c}}_1$, then the values of (A, B) is

• $\left(\sin \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\alpha }\right)\right)$

• $\left(\cos \left(\mathrm{\alpha }\right), \sin \left(\mathrm{\alpha }\right)\right)$

• $\left(- \sin \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\alpha }\right)\right)$

• $\left(- \cos \left(\mathrm{\alpha }\right), \sin \left(\mathrm{\alpha }\right)\right)$