$\mathrm{If} \int {\mathrm{x}}^3{\mathrm{e}}^{5\mathrm{x}}\mathrm{dx} = \frac{{\mathrm{e}}^{5\mathrm{x}}}{5^4}\left[\mathrm{f}\left(\mathrm{x}\right)\right] + \mathrm{C},\mathrm{then} \mathrm{f}\left(\mathrm{x}\right) = ?$

• $\frac{{\mathrm{x}}^3}{5} - \frac{3{\mathrm{x}}^2}{5^2} + \frac{6\mathrm{x}}{5^3} - \frac{6}{5^4}$

• $5{\mathrm{x}}^3 - 5^2{\mathrm{x}}^2 + 5^3\mathrm{x} - 6$

• $5^3{\mathrm{x}}^3 - 15{\mathrm{x}}^2 + 30\mathrm{x} - 6$

• $5^3{\mathrm{x}}^3 - 75{\mathrm{x}}^2 + 30\mathrm{x} - 6$

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

• $\frac{{\mathrm{x}}^2 + 2}{{\mathrm{x}}^2 +\hspace{0.17em}2\mathrm{x} + 2} - \frac{1}{2}{\tan }^{-1}\left(\mathrm{x} - 1\right) + \mathrm{C}$

• $\frac{{\mathrm{x}}^2 - 2}{4\left({\mathrm{x}}^2 + 2\mathrm{x} +\hspace{0.17em}2\right)} - \frac{1}{2}{\tan }^{-1}\left(\mathrm{x} +\hspace{0.17em}1\right) +\hspace{0.17em}\mathrm{C}$

• $\frac{{\mathrm{x}}^2 + 2}{2\left({\mathrm{x}}^2 + 2\mathrm{x} +\hspace{0.17em}2\right)} - \frac{1}{2}{\tan }^{-1}\left(\mathrm{x} +\hspace{0.17em}1\right) +\hspace{0.17em}\mathrm{C}$

• $\frac{2\left(\mathrm{x} - 1\right)}{{\mathrm{x}}^2 +\hspace{0.17em}2\mathrm{x} +\hspace{0.17em}2} + \frac{1}{2}{\tan }^{-1}\left(\mathrm{x} +\hspace{0.17em}1\right) +\hspace{0.17em}\mathrm{C}$

$\mathrm{If} \int \log \left({\mathrm{a}}^2 + {\mathrm{x}}^2\right)\mathrm{dx} = \mathrm{h}\left(\mathrm{x}\right) + \mathrm{C}, \mathrm{then} \mathrm{h}\left(\mathrm{x}\right) = ?$

• $\mathrm{xlog}\left({\mathrm{a}}^2 +\hspace{0.17em}{\mathrm{x}}^2\right) + 2{\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)$

• ${\mathrm{x}}^2\log \left({\mathrm{a}}^2 +\hspace{0.17em}{\mathrm{x}}^2\right) +\hspace{0.17em}\mathrm{x} +\hspace{0.17em}{\mathrm{atan}}^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)$

• $\mathrm{xlog}\left({\mathrm{a}}^2 + {\mathrm{x}}^2\right) - 2\mathrm{x} + 2{\mathrm{atan}}^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)$

• ${\mathrm{x}}^2\log \left({\mathrm{a}}^2 +\hspace{0.17em}{\mathrm{x}}^2\right) + 2\mathrm{x} - {\mathrm{a}}^2{\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)$

$$\begin{gathered} \mathrm{For} \mathrm{x} > 0, \mathrm{if} \int \left(\log {\left(\mathrm{x}\right)}^5\right)\mathrm{dx} = ? \\ \mathrm{x}\left[\mathrm{A}\left(\log {\left(\mathrm{x}\right)}^5\right) + \mathrm{B}{\left(\log \left(\mathrm{x}\right)\right)}^4+\hspace{0.17em}\mathrm{C}{\left(\log \left(\mathrm{x}\right)\right)}^3 + \mathrm{D}{\left(\log \left(\mathrm{x}\right)\right)}^2 +\hspace{0.17em}\mathrm{E}\left(\log \left(\mathrm{x}\right)\right) + \mathrm{F}\right] + \mathrm{Constant}, \mathrm{then} \\ \mathrm{A} +\hspace{0.17em}\mathrm{B} +\hspace{0.17em}\mathrm{C} +\hspace{0.17em}\mathrm{D}\hspace{0.17em}+\hspace{0.17em}\mathrm{E}\hspace{0.17em}+ \mathrm{F} = ? \end{gathered}$$

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By the definition of the definite integral, the value of $\underset{\mathrm{n}\to \infty }{\lim }\left(\frac{1}{\sqrt{{\mathrm{n}}^2 - 1}} + \frac{{\displaystyle 1}}{{\displaystyle \sqrt{{\mathrm{n}}^2 - 2^2}}} + ... \frac{{\displaystyle 1}}{{\displaystyle \sqrt{{\mathrm{n}}^2 - {\left(\mathrm{n} - 1\right)}^2}}}\right)$ is equal to

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

${\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{4}}\left(\frac{\mathrm{x} + \frac{\mathrm{\pi }}{4}}{2 - \cos \left(2\mathrm{x}\right)}\right)\mathrm{dx}$ is equal to

• $\frac{8\mathrm{\pi }\sqrt{3}}{5}$

• $\frac{2\mathrm{\pi }\sqrt{3}}{9}$

• $\frac{4{\mathrm{\pi }}^2\sqrt{3}}{9}$

• $\frac{{\mathrm{\pi }}^2}{6\sqrt{3}}$

$\int \frac{5\mathrm{x} + 3}{{\mathrm{x}}^2\left({\mathrm{x}}^2 - 2\right)}\mathrm{dx} = ?$

• $\frac{3}{2\mathrm{x}} + \frac{\sqrt{13}}{2\sqrt{2}}\log \left(\left|\frac{\sqrt{2} - \mathrm{x}}{\sqrt{2} + \mathrm{x}}\right|\right) + \mathrm{C}$

• $\frac{3}{2\mathrm{x}} + \frac{13}{4\sqrt{2}}\log \left(\left|\frac{x + \sqrt{2}}{x - \sqrt{2}}\right|\right) + \mathrm{C}$

• $\frac{3}{2\mathrm{x}} + \frac{13}{4\sqrt{2}}\log \left(\left|\frac{\mathrm{x} - \sqrt{2}}{\mathrm{x} + \sqrt{2}}\right|\right) + \mathrm{C}$

• $\frac{3}{5}x + \frac{5}{3\sqrt{2}}\log \left(\left|\frac{x + \sqrt{2}}{x - \sqrt{2}}\right|\right) + \mathrm{C}$

$\mathrm{If} \mathrm{y} = {\tan }^{-1}\left(\left\{\frac{\mathrm{x}}{1 + \sqrt{1 - {\mathrm{x}}^2}}\right\}\right) + \sin \left(2{\tan }^{-1}\left(\sqrt{\frac{1 - \mathrm{x}}{1 + \mathrm{x}}}\right)\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

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

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

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

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

If ${\int }_0^{10}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}= 5, \mathrm{then} \sum _{\mathrm{k} = 1}^{10}{\int }_0^1\mathrm{f}\left(\mathrm{k} - 1 + \mathrm{x}\right)\mathrm{dx}$ = ?

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$\int \frac{{\mathrm{x}}^{\mathrm{e} - 1} + {\mathrm{e}}^{\mathrm{x} - 1}}{{\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}}\mathrm{dx}$

• $\frac{- 1}{\mathrm{e}}\log \left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

• $- \mathrm{elog}\left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

• $\frac{1}{\mathrm{e}}\log \left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

• $\mathrm{e}\log \left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$