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

• 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

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

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

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

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

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

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

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

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

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

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

956.

$\mathrm{If} \int \left(1 - \cos \left(\mathrm{x}\right)\right){\csc }^2\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c}, \mathrm{then} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\tan \left(\frac{\mathrm{x}}{2}\right)$

• $\cot \left(\frac{\mathrm{x}}{2}\right)$

• $2\tan \left(\frac{\mathrm{x}}{2}\right)$

• $\frac{1}{2}\tan \left(\frac{\mathrm{x}}{2}\right)$

$$\begin{gathered} \mathrm{If} {\mathrm{I}}_{\mathrm{n}} = {\int }_0^{\raisebox{1ex}{${\mathrm{n}}^{}\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\tan \left(\mathrm{x}\right)\mathrm{dx}, \mathrm{then} \\ {\mathrm{I}}_2 + {\mathrm{I}}_4, {\mathrm{I}}_3 + {\mathrm{I}}_5, {\mathrm{I}}_4 + {\mathrm{I}}_6, . . . , \mathrm{are} \mathrm{in} \end{gathered}$$

• arithmatic progression

• geometric progression

• harmonic progression

• arithmetic-geometric progression

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

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

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

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

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

$\mathrm{If} \int \frac{{\sin }^8\left(\mathrm{x}\right) - {\cos }^8\left(\mathrm{x}\right)}{1 - 2{\sin }^2\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)}\mathrm{dx} = \mathrm{Asin}\left(2\mathrm{x}\right) + \mathrm{B}, \mathrm{then} \mathrm{A} = ?$

• $- \frac{1}{2}$

• - 1

• $\frac{1}{2}$

• 1

$\int \frac{1 + \cos \left(4\mathrm{x}\right)}{\cot \left(\mathrm{x}\right) - \tan \left(\mathrm{x}\right)}\mathrm{dx}$

• $- \frac{1}{4}\cos \left(4\mathrm{x}\right) + \mathrm{C}$

• $\frac{1}{8}\cos \left(4\mathrm{x}\right) + \mathrm{C}$

• $\frac{1}{4}\sin \left(4\mathrm{x}\right) + \mathrm{C}$

• $- \frac{1}{8}\cos \left(4\mathrm{x}\right) + \mathrm{C}$