Mathematics : Integrals

$\mathrm{If} {\mathrm{I}}_{\mathrm{n}} = {\int }_0^{\raisebox{1ex}{$\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

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

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

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

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

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

$\mathrm{If} \mathrm{a} > 0, \mathrm{then} {\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}\frac{{\sin }^2\left(\mathrm{x}\right)}{1 + {\mathrm{a}}^{\mathrm{x}}}\mathrm{dx} = ?$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $\frac{2\mathrm{\pi }}{2}$

• a$\mathrm{\pi }$

$$\begin{gathered} \mathrm{The} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{integral} {\int }_0^4\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2} \mathrm{obtained} \mathrm{by} \mathrm{using} \mathrm{trapezoidal} \\ \mathrm{rule} \mathrm{with} \mathrm{h} = 1 \mathrm{is} \end{gathered}$$

• $\frac{63}{85}$

• ${\tan }^{-1}\left(4\right)$

• $\frac{108}{85}$

• $\frac{113}{85}$

$$\begin{gathered} \mathrm{Let} \mathrm{A} = \left|\begin{array}{cc}2& {\mathrm{e}}^{\mathrm{i\pi }}\\ - 1& {\mathrm{i}}^{2012}\end{array}\right|, \mathrm{C}= \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{x}}\right)\mathrm{x} = 1, \\ \mathrm{D} = {\int }_{{\mathrm{e}}^2}^1\frac{\mathrm{dx}}{\mathrm{x}}. \mathrm{If} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{two} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{Ax}}^3 + {\mathrm{Bx}}^2 + \mathrm{Cx} - \mathrm{D} = 0 \\ \mathrm{is} \mathrm{equal} \mathrm{to} \mathrm{zero}, \mathrm{then} \mathrm{B} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• - 1

• 0

• 1

• 2

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

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

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

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

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

$\mathrm{If} \int \frac{\mathrm{x} - \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx} = \mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \mathrm{plog}\left|\sec \left(\frac{\mathrm{x}}{2}\right)\right| + \mathrm{C}, \mathrm{then} \mathrm{p} = ?$

• - 4

• 4

• 2

• - 2