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

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

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

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

$\int \frac{4^{\mathrm{x} + 1} - 7^{\mathrm{x} - 1}}{{28}^{\mathrm{x}}}\mathrm{dx}$ is equal to

• $\frac{1}{7{\log }_{\mathrm{e}}\left(4\right)}{4}^{- \mathrm{x}} - \frac{4}{{\log }_{\mathrm{e}}\left(7\right)}{7}^{- \mathrm{x}} + \mathrm{C}$

• $\frac{1}{7{\log }_{\mathrm{e}}\left(4\right)}{4}^{- \mathrm{x}} + \frac{4}{{\log }_{\mathrm{e}}\left(7\right)}{7}^{- \mathrm{x}} + \mathrm{C}$

• $\frac{4^{- \mathrm{x}}}{{\log }_{\mathrm{e}}\left(7\right)} - \frac{7^{- \mathrm{x}}}{{\log }_{\mathrm{e}}\left(4\right)} + \mathrm{C}$

• $\frac{4^{- \mathrm{x}}}{{\log }_{\mathrm{e}}\left(4\right)} - \frac{7^{- \mathrm{x}}}{{\log }_{\mathrm{e}}\left(7\right)} + \mathrm{C}$

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

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

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

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

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

148.

The value of $\int \frac{{\mathrm{e}}^{5{\log }_{\mathrm{e}}\left(\mathrm{x}\right)} - {\mathrm{e}}^{4{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}}{{\mathrm{e}}^{3{\log }_{\mathrm{e}}\left(\mathrm{x}\right)} - {\mathrm{e}}^{2{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}}\mathrm{dx}$ is

• x² + C

• $\frac{{\mathrm{x}}^2}{2} + \mathrm{C}$

• $\frac{{\mathrm{x}}^3}{3} + \mathrm{C}$

• $\frac{\mathrm{x}}{2} + \mathrm{C}$

If $\mathrm{f}\left(\mathrm{x}\right) = \frac{{\sin }^{-1}\left(\mathrm{x}\right)}{\sqrt{1 - {\mathrm{x}}^2}} \mathrm{and} \mathrm{g}\left(\mathrm{x}\right) = {\mathrm{e}}^{{\sin }^{-1}\left(\mathrm{x}\right)}$, then $\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

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

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

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

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

The value of $\int \frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^4 - {\mathrm{x}}^2 + 1}\mathrm{dx}$ is

• ${\tan }^{-1}\left(2{\mathrm{x}}^2 - 1\right) + \mathrm{C}$

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

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

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