If $\int {\sin }^{-1}\left(\frac{2\mathrm{x}}{1 + {\mathrm{x}}^2}\right)\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) - \log \left(1 + {\mathrm{x}}^2\right)$ then f(x) is equal to

• 2xtan^-1(x)

• - 2xtan^-1(x)

• xtan^-1(x)

• - xtan^-1(x)

$\mathrm{If} \int \sqrt{\frac{\mathrm{x}}{{\mathrm{a}}^3 - {\mathrm{x}}^3}}\mathrm{dx} = \mathrm{g}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}, \mathrm{then} \mathrm{g}\left(\mathrm{x}\right) \mathrm{is} \mathrm{equal} \mathrm{to} :$

• $\frac{2}{3}{\cos }^{-1}\left(\mathrm{x}\right)$

• $\frac{2}{3}{\sin }^{-1}\left(\frac{{\mathrm{x}}^3}{{\mathrm{a}}^3}\right)$

• $\frac{2}{3}{\sin }^{-1}\left(\sqrt{\frac{{\mathrm{x}}^3}{{\mathrm{a}}^3}}\right)$

• $\frac{2}{3}{\cos }^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)$

If $\int \frac{\mathrm{dx}}{{\mathrm{x}}^2 + 2\mathrm{x} + 2} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c}$, then f(x) is equal to

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

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

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

• $3{\tan }^{-1}\left(\mathrm{x} + 1\right)$

Observe the following statement:

$$\begin{gathered} \mathrm{A} : \int \left(\frac{{\mathrm{x}}^2 - 1}{{\mathrm{x}}^2}\right){\mathrm{e}}^{\frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2}}\mathrm{dx} = {\mathrm{e}}^{\frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2}} + \mathrm{c} \\ \mathrm{R} : \int \mathrm{f}\text{'}\left(\mathrm{x}\right){\mathrm{e}}^{\mathrm{f}\left(\mathrm{x}\right)}\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c} \end{gathered}$$

Then which of the followmg is true ?

• Both A and R are true and R is not thecorrect reason for A

• Both A and R are true and R is the correct reason for A

• A is true, R is false

• A is false, R is true

Dividing the interval [0, 6] into 6 equal parts and by using trapezoidal rule the value of ${\int }_0^6{\mathrm{x}}^3\mathrm{dx}$ is approximately

• 330

• 331

• 332

• 333

936.

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \frac{\mathrm{dx}}{1 + {\tan }^3\left(\mathrm{x}\right)} = ?$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{3\mathrm{\pi }}{2}$

${\int }_{- 1}^1 \frac{\cos \left(\mathrm{hx}\right)}{1 +\hspace{0.17em}{\mathrm{e}}^{2\mathrm{x}}}d\mathrm{x} = ?$

• 0

• 1

• $\frac{{\mathrm{e}}^2 - 1}{2\mathrm{e}}$

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

$\mathrm{If} \int \frac{{\mathrm{e}}^{\mathrm{x}} - 1}{{\mathrm{e}}^{\mathrm{x}} + 1}\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c}, \mathrm{then} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

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

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

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

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

$\int {\tan }^{-1}\left(\sqrt{\frac{1 - \mathrm{x}}{1 + \mathrm{x}}}\right)\mathrm{dx} = ?$

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

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

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

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

$\frac{\sin \left(\mathrm{x}\right) + 8\cos \left(\mathrm{x}\right)}{4\sin \left(\mathrm{x}\right) + 6\cos \left(\mathrm{x}\right)}\mathrm{dx} = ?$

• $\mathrm{x} + \frac{1}{2}\log \left(4\sin \left(\mathrm{x}\right) + 6\cos \left(\mathrm{x}\right)\right) + \mathrm{c}$

• $2\mathrm{x} + \log \left(2\sin \left(\mathrm{x}\right) + 3\cos \left(\mathrm{x}\right)\right) + \mathrm{c}$

• $\mathrm{x} + \log \left(2\sin \left(\mathrm{x}\right) + 3\cos \left(\mathrm{x}\right)\right) + \mathrm{c}$

• $\frac{1}{2}\log \left(4\sin \left(\mathrm{x}\right) + 6\cos \left(\mathrm{x}\right)\right) + \mathrm{c}$