671.

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

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

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

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

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

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

• 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

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

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

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

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

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

$\mathrm{If} {\mathrm{I}}_{\mathrm{n}} = {\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\tan }^{\mathrm{n}}\left(\mathrm{d\theta }\right) \mathrm{for} \mathrm{n} = 1, 2, 3 . . . , \mathrm{then} {\mathrm{I}}_{\mathrm{n} + 1} + {\mathrm{I}}_{ \mathrm{n} + 1} = ?$

• 0

• 1

• $\frac{1}{\mathrm{n}}$

• $\frac{1}{\mathrm{n} + 1}$

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(0\right) = 1, \mathrm{f}\left(0.5\right) = \frac{5}{4}, \mathrm{f}\left(1\right) = 2, \mathrm{f}\left(1.5\right) = \frac{13}{4}, \mathrm{and} \mathrm{f}\left(2\right) = 5. \mathrm{Using} \mathrm{Simson}\text{'}\mathrm{s} \mathrm{rule}, \\ {\int }_0^2\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = ? \end{gathered}$$

• $\frac{14}{3}$

• $\frac{7}{6}$

• $\frac{14}{9}$

• $\frac{7}{9}$

$\int \frac{\mathrm{dx}}{{\mathrm{x}}^2\sqrt{4 + {\mathrm{x}}^2}} =$

• $\frac{1}{4}\sqrt{4 + {\mathrm{x}}^2} + \mathrm{C}$

• $- \frac{1}{4}\sqrt{4 + {\mathrm{x}}^2} + \mathrm{C}$

• $- \frac{1}{4\mathrm{x}}\sqrt{4 + {\mathrm{x}}^2} + \mathrm{C}$

• $\frac{9}{4\mathrm{x}}\sqrt{4 + {\mathrm{x}}^2} + \mathrm{C}$

$\int {\sec }^2\left(\mathrm{x}\right){\csc }^4\left(\mathrm{x}\right)\mathrm{dx} = ?$

• 4

• 3

• 2

• 1

$\int \frac{\mathrm{dx}}{\sqrt{\mathrm{x} - {\mathrm{x}}^2}} = ?$

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

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

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

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