$\int \left(1 + \mathrm{x} - {\mathrm{x}}^{- 1}\right){\mathrm{e}}^{\mathrm{x} + {\mathrm{x}}^{- 1}}\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to}˸$

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

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

• $- {\mathrm{xe}}^{\mathrm{x} + {\mathrm{x}}^{- 1}} + \mathrm{C}$

• ${\mathrm{xe}}^{\mathrm{x} + {\mathrm{x}}^{- 1}} + \mathrm{C}$

${\int }_{- 2}^2\left|\begin{array}{c}\left[\mathrm{x}\right]\end{array}\right|\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to}$

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${\int }_0^1\sin \left(2{\tan }^{-1}\left(\sqrt{\frac{1 + \mathrm{x}}{1 - \mathrm{x}}}\right)\right)\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

${\int }_0^3\frac{3\mathrm{x} + 1}{{\mathrm{x}}^2 + 9}\mathrm{dx}$ is equal to :

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{12}$

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{2}$

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{6}$

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{3}$

If [2, 6] is divided into four intervals of equal length, then the approximate value of ${\int }_2^6\frac{1}{{\mathrm{x}}^2 - \mathrm{x}}\mathrm{dx}$ using Simpson's rule, is

• 0.3222

• 0.2333

• 0.5222

• 0.2555

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \frac{1}{{\mathrm{x}}^2}{\int }_3^{\mathrm{x}}\left(2\mathrm{t} - 3\mathrm{f}\text{'}\left(\mathrm{t}\right)\right)\mathrm{dt}, \mathrm{then} \mathrm{f}\text{'}\left(\mathrm{t}\right), \mathrm{then} \mathrm{f}\text{'}\left(3\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{- 1 }{2}$

• $\frac{- 1}{3}$

• $\frac{1}{2}$

• $\frac{1}{3}$

$\int \frac{\mathrm{dx}}{\left(\mathrm{x} + 100\right)\sqrt{\mathrm{x} + 99}} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c} \Rightarrow \mathrm{f}\left(\mathrm{x}\right)$

• 2(x + 100)^1/2

• 3(x + 100)^1/2

• $2{\tan }^{-1}\left(\sqrt{\mathrm{x} + 99}\right)$

• $2{\tan }^{-1}\left(\sqrt{\mathrm{x} + 100}\right)$

$\int \frac{3 - {\mathrm{x}}^2}{1 - 2\mathrm{x} + {\mathrm{x}}^2}{\mathrm{e}}^{\mathrm{x}}\mathrm{dx} = {\mathrm{e}}^{\mathrm{x}}\mathrm{f}\left(\mathrm{x}\right) + \mathrm{c} \Rightarrow \mathrm{f}\left(\mathrm{x}\right)$

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

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

• $\frac{1 - \mathrm{x}}{\mathrm{x} - 1}$

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

$\int \frac{\sqrt{\cot \left(\mathrm{x}\right)}}{\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}\mathrm{dx} = - \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c} \Rightarrow \mathrm{f}\left(\mathrm{x}\right)$

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

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

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

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

${\int }_{\frac{- \mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\log \left(\frac{2 - \sin \left(\mathrm{\theta }\right)}{2 + \sin \left(\mathrm{\theta }\right)}\right)\mathrm{d\theta }$ is equal to

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• - 1