${\int }_0^1{\mathrm{x}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{1 - \mathrm{x}}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{9}$

• $\frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{16}$

${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left|\mathrm{x}\right|\mathrm{dx}$ is equal to

• 0

• 1

• 2

• $\mathrm{\pi }$

$\int \frac{\mathrm{dx}}{\left(\mathrm{x} + 1\right)\sqrt{4\mathrm{x} + 3}} = ?$

• ${\text{tan}}^{-1}\left(\sqrt{4x + 3}\right) + c$

• $3{\tan }^{-1}\left(\sqrt{4\mathrm{x} + 3}\right) +\hspace{0.17em}\mathrm{c}$

• $2{\tan }^{-1}\left(\sqrt{4\mathrm{x} + 3}\right) + \mathrm{c}$

• $4{\tan }^{-1}\left(\sqrt{4\mathrm{x} + 3}\right) + \mathrm{c}$

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

• $- \cot \left(\mathrm{x}\right) {\mathrm{e}}^{\mathrm{x}} +\hspace{0.17em}\mathrm{c}$

• $cot\left(x\right) e^x +\hspace{0.17em}c$

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

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

665.

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

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

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

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

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

${\int }_0^{\mathrm{\pi }}\frac{1}{1 + \sin \left(\mathrm{x}\right)}d\mathrm{x} =?$

• 1

• 2

• - 1

• - 2

$$\begin{gathered} \mathrm{The} \mathrm{line} \mathrm{x} = \frac{\mathrm{\pi }}{4} \mathrm{divides} \mathrm{the} \mathrm{area} \mathrm{of} \mathrm{the} \mathrm{region} \mathrm{bounded} \mathrm{by} \mathrm{y} = \sin \left(\mathrm{x}\right), \mathrm{y} = \cos \left(\mathrm{x}\right) \\ \mathrm{and} \mathrm{x}-\mathrm{axis} \left(0 \le \mathrm{x} \frac{\mathrm{\pi }}{2}\right) \mathrm{into} \mathrm{two} \mathrm{regions} \mathrm{of} \mathrm{areas} {\mathrm{A}}_1 \mathrm{and} {\mathrm{A}}_2. \mathrm{Then} {\mathrm{A}}_1, {\mathrm{A}}_2 \mathrm{equals} \end{gathered}$$

• 4 : 1

   

• 3 : 1

• 2 : 1

• 1 : 1

The velocity of a particle which starts from rest is given by the following table

t (in sec)	0	2	4	6	8	10
v (in m/s)	0	12	16	20	35	60

The total distance travelled (in metre) by the particles in 10s, using trapezoidal rule is given by

• 113

• 226

• 143

• 246

$\mathrm{If} \int \frac{7{\mathrm{x}}^8 + 8{\mathrm{x}}^7}{{\left(1 + \mathrm{x} + {\mathrm{x}}^8\right)}^2}\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c}, \mathrm{then} \mathrm{f}\left(\mathrm{x}\right) = ?$

• $\frac{{\mathrm{x}}^8}{1 + \mathrm{x} + {\mathrm{x}}^8}$

• $28\log \left(1 + \mathrm{x} + {\mathrm{x}}^8\right)$

• $\frac{1}{1 + \mathrm{x} + {\mathrm{x}}^8}$

• $\frac{- 1}{1 + \mathrm{x} + {\mathrm{x}}^8}$

$$\begin{gathered} \mathrm{If} {\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right) = \log \log \log . . .\log \left(\mathrm{x}\right) \left(\log \mathrm{is} \mathrm{repeated} \mathrm{n}-\mathrm{times}\right), \mathrm{then} \\ \int {\left({\mathrm{xf}}_1\left(\mathrm{x}\right){\mathrm{f}}_2\left(\mathrm{x}\right) . . . {\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right)\right)}^{ - 1}\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• ${\mathrm{f}}_{\mathrm{n} + 1} + \mathrm{c}$

• $\frac{{\mathrm{f}}_{\mathrm{n} + 1}\left(\mathrm{x}\right)}{\mathrm{n} + 1} + \mathrm{c}$

• ${\mathrm{nf}}_{\mathrm{n}\left(\mathrm{x}\right) }+ \mathrm{c}$

• $\frac{{\mathrm{f}}_{\mathrm{n}}\left(\mathrm{x}\right)}{\mathrm{n}} + \mathrm{c}$