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

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $\frac{2\mathrm{\pi }}{2}$

• 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

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

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

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

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

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

685.

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

• - 4

• 4

• 2

• - 2

$\mathrm{If} \int \frac{\mathrm{dx}}{\mathrm{xlog}\left(\mathrm{x} - 2\right)\log \left(\mathrm{x} - 3\right)} = \mathrm{I} + \mathrm{C}, \mathrm{then} \mathrm{I} =?$

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

• $\log \left|\frac{\log \left(\mathrm{x} - 3\right)}{\log \left(\mathrm{x} - 2\right)}\right|$

• $\log \left|\frac{\log \left(\mathrm{x} - 2\right)}{\log \left(\mathrm{x} - 3\right)}\right|$

• $\log \left|\log \left(\mathrm{x} - 3\right)\log \left(\mathrm{x} - 2\right)\right|$

$\mathrm{If} {\int }_0^{\mathrm{b}}\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2} = {\int }_{\mathrm{b}}^{\infty }\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}, \mathrm{then} \mathrm{b} = ?$

• ${\tan }^{-1}\left(\frac{1}{3}\right)$

• $\frac{\sqrt{3}}{2}$

• $\sqrt{2}$

• 1

The approximate value of ${\int }_1^3\frac{\mathrm{dx}}{2 + 3\mathrm{x}}$ using Simpson's rule and dividing the interval [1, 3] into two equal parts is

• $\frac{1}{3}\log \left(\frac{11}{5}\right)$

• $\frac{107}{110}$

• $\frac{22}{110}$

• $\frac{119}{440}$

$\mathrm{If} \int \frac{\mathrm{dx}}{\sqrt{{\sin }^3\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}} = \mathrm{g}\left(\mathrm{x}\right) + \mathrm{c}, \mathrm{then} \mathrm{g}\left(\mathrm{x}\right) = ?$

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

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

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

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

$\mathrm{If} \int \frac{\mathrm{dx}}{\left(1 + \sqrt{\mathrm{x}}\right)\sqrt{\mathrm{x} - {\mathrm{x}}^2}} = \frac{\mathrm{A}\sqrt{\mathrm{x}}}{\sqrt{1 -\mathrm{x}}} + \frac{\mathrm{B}}{\sqrt{1 - \mathrm{x}}} + \mathrm{C}, \mathrm{where} \mathrm{C} \mathrm{is} \mathrm{real} \mathrm{constant}, \mathrm{then} \mathrm{A} + \mathrm{B} = ?$

• 3

• 0

• 1

• 2