$$\begin{gathered} \mathrm{If} \int \left({\mathrm{e}}^{2\mathrm{x}} + 2{\mathrm{e}}^{\mathrm{x}} - {\mathrm{e}}^{ - \mathrm{x}} - 1\right){\mathrm{e}}^{\left({\mathrm{e}}^{\mathrm{x}} +{\mathrm{e}}^{ - \mathrm{x}}\right)}\mathrm{dx} = \mathrm{g}\left(\mathrm{x}\right){\mathrm{e}}^{\left({\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{ - \mathrm{x}}\right)} +\hspace{0.17em}\mathrm{C}, \\ \mathrm{where} \mathrm{c} \mathrm{is} \mathrm{a} \mathrm{constant} \mathrm{of} \mathrm{integration}, \mathrm{then} \mathrm{g}\left(0\right) \mathrm{is} \end{gathered}$$

• 1

• e

• e²

• 2

The value of ${\int }_{\raisebox{1ex}{$ - \mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{1}{1 +{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}d\mathrm{x} \mathrm{is} :$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

$\mathrm{If} \int \frac{\cos \left(\mathrm{\theta }\right)}{5 + 7\sin \left(\mathrm{\theta }\right) - 2{\cos }^2\left(\mathrm{\theta }\right)}\mathrm{d\theta } = {\mathrm{Alog}}_{\mathrm{e}}\left|\mathrm{B}\left(\mathrm{\theta }\right)\right| + \mathrm{C}, \mathrm{where} \mathrm{C} \mathrm{is} \mathrm{a} \mathrm{constant} \mathrm{of} \mathrm{integration}, \mathrm{then} \frac{\mathrm{B}\left(\mathrm{\theta }\right)}{\mathrm{A}}\mathrm{can} \mathrm{be}:$

• $\frac{2\sin \left(\mathrm{\theta }\right) + 1}{5\left(\sin \left(\mathrm{\theta }\right) + 3\right)}$

• $\frac{5\left(2\sin \left(\mathrm{\theta }\right) + 1\right)}{\sin \left(\mathrm{\theta }\right) + 3}$

• $\frac{2\sin \left(\mathrm{\theta }\right) + 1}{\sin \left(\mathrm{\theta }\right) + 3}$

• $\frac{5\left(\sin \left(\mathrm{\theta }\right) + 3\right)}{2\sin \left(\mathrm{\theta }\right) + 1}$

The general solution of the differential equation

$\sqrt{1 + {\mathrm{x}}^2 + {\mathrm{y}}^2 +{\mathrm{x}}^2{\mathrm{y}}^2} + \mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0 \left(\mathrm{where} \mathrm{C} \mathrm{is} \mathrm{constant} \mathrm{of} \mathrm{integration} \right)$

• $\sqrt{1 + {\mathrm{y}}^2} + \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 + {\mathrm{x}}^2} - 1}{\sqrt{1 + {\mathrm{x}}^2 +1}}\right) +\mathrm{C}$

• $\sqrt{1 + {\mathrm{y}}^2} + \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 \hspace{0.17em}+ {\mathrm{x}}^2} +\hspace{0.17em}1}{\sqrt{1 +{\mathrm{x}}^2} - 1}\right) +\hspace{0.17em}\mathrm{C}$

• $\sqrt{1 + {\mathrm{y}}^2} - \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 \hspace{0.17em}+ {\mathrm{x}}^2} -\hspace{0.17em}1}{\sqrt{1 +{\mathrm{x}}^2 + 1} }\right) +\hspace{0.17em}\mathrm{C}$

• $\sqrt{1 + {\mathrm{y}}^2} - \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 \hspace{0.17em}+ {\mathrm{x}}^2} +\hspace{0.17em}1}{\sqrt{1 +{\mathrm{x}}^2 - 1} }\right) +\hspace{0.17em}\mathrm{C}$

$\mathrm{If} {\mathrm{I}}_1 = {\int }_0^1{\left(1 - {\mathrm{x}}^{50}\right)}^{100}d\mathrm{x} \mathrm{and} {\mathrm{I}}_2 = {\int }_0^1{\left(1 - {\mathrm{x}}^{50}\right)}^{101}d\mathrm{x} {\mathrm{I}}_2 = {\mathrm{\alpha I}}_1 \mathrm{Then} \mathrm{\alpha } = ?$

• $\frac{5049}{5050}$

• $\frac{5051}{5050}$

• $\frac{5050}{5051}$

• $\frac{5050}{5049}$

$\mathrm{The} \mathrm{integral} {\int }_1^2{\mathrm{e}}^{\mathrm{x}} . {\mathrm{x}}^2\left(2 + {\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)d\mathrm{x} = ?$

• e(2e - 1)

• e(4e + 1)

• 4e² - 1

• e(4e - 1)

The common difference of the A.P. b₁,b₂,....,bm is 2 more than common difference of A.P. a₁,a₂,...,a_n. If a₄₀ = –159, a₁₀₀ = – 399 and b₁₀₀ = a₇₀, then b₁is equal to : 

• 127

• 81

• - 127

• - 81