Mathematics : Integrals

If ${\int }_1^{\mathrm{x}}\frac{d\mathrm{t}}{\left|\mathrm{t}\right|\sqrt{{\mathrm{t}}^2 - 1}} = \frac{\mathrm{\pi }}{6}$, then x can be equal to

• $\frac{2}{\sqrt{3}}$

• $\sqrt{3}$

• 2

• None of these

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

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

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

• $2{\mathrm{e}}^{\mathrm{x}}\cot \left(\mathrm{x}\right) + \mathrm{c}$

• $- 2{\mathrm{e}}^{\mathrm{x}}\cot \left(\mathrm{x}\right) + \mathrm{c}$

$\mathrm{If} {\mathrm{I}}_{\mathrm{n}} = \int {\sin }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}, \mathrm{then} {\mathrm{nI}}_{\mathrm{n}} - \left(\mathrm{n} - 1\right){\mathrm{I}}_{\mathrm{n} - 2}$ equals

• ${\sin }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\mathrm{xcos}\left(\mathrm{x}\right)$

• ${\cos }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)$

• $- {\sin }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)$

• $- {\cos }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)$

If I = $\int \frac{{\mathrm{x}}^5}{\sqrt{1 + {\mathrm{x}}^3}}\mathrm{dx}$, then I is equal to

• $\frac{2}{9}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{5}{2}} + \frac{2}{3}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{3}{2}} + \hspace{0.17em}\mathrm{c}$

• $\log \left|\sqrt{\mathrm{x}} + \sqrt{1 + {\mathrm{x}}^3}\right| + \mathrm{c}$

• $\log \left|\sqrt{\mathrm{x}} - \sqrt{1 + {\mathrm{x}}^3}\right| + \mathrm{c}$

• $\frac{2}{9}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{3}{2}} - \frac{2}{3}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{1}{2}} + \hspace{0.17em}\mathrm{c}$

The value of ${\int }_0^{\mathrm{a}}\sqrt{\frac{\mathrm{a} - \mathrm{x}}{\mathrm{x}}}d\mathrm{x}$ is

• $\frac{\mathrm{a}}{2}$

• $\frac{\mathrm{a}}{4}$

• $\frac{\mathrm{\pi a}}{2}$

• $\frac{\mathrm{\pi a}}{4}$

The value of ${\int }_1^4\left|\mathrm{x} - 3\right|\mathrm{dx}$ is equal to :

• 1

• $\frac{5}{2}$

• $\frac{1}{2}$

• $\frac{3}{2}$

The value of ${\int }_0^{\infty }\frac{\mathrm{dx}}{\left({\mathrm{a}}^2 + {\mathrm{x}}^2\right)}$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2\mathrm{a}}$

• $\frac{\mathrm{\pi }}{\mathrm{a}}$

• $\frac{1}{2\mathrm{a}}$

The value of the integral $\int {\mathrm{e}}^{\mathrm{x}}{\left(\frac{1 - \mathrm{x}}{1 + {\mathrm{x}}^2}\right)}^2\mathrm{dx}$ is

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{1 - \mathrm{x}}{1 + {\mathrm{x}}^2}\right) + \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{1 + \mathrm{x}}{1 + {\mathrm{x}}^2}\right)$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{1 + {\mathrm{x}}^2} + \mathrm{c}$

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

If g(x) = ${\int }_0^{\mathrm{x}}{\cos }^4\left(\mathrm{t}\right)d\mathrm{t}$, then $\mathrm{g}\left(\mathrm{x} + \mathrm{\pi }\right)$ is equsl to

• g(x) + g($\mathrm{\pi }$)

• g(x) - g()

• g(x) . g($\mathrm{\pi }$)

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

$\int \left({\sin }^6\left(\mathrm{x}\right) + {\cos }^6\left(\mathrm{x}\right) + 3{\sin }^2\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to

• x + C

• $\frac{3}{2}\sin \left(2\mathrm{x}\right) + \mathrm{C}$

• $- \frac{3}{2}\cos \left(2\mathrm{x}\right) + \mathrm{C}$

• $\frac{1}{3}\sin \left(3\mathrm{x}\right) - \cos \left(3\mathrm{x}\right) + \mathrm{C}$