What is $\int \frac{\mathrm{dx}}{\mathrm{x}{\left(1 + \ln \left(\mathrm{x}\right)\right)}^{\mathrm{n}}}$ equal to (n ≠ 1) ?

• $\frac{1}{\left(\mathrm{n} - 1\right){\left(1 + \ln \left(\mathrm{x}\right)\right)}^{\mathrm{n} - 1}} +\hspace{0.17em}\mathrm{c}$

• $\frac{1 - \mathrm{n}}{{\left(1 + \ln \left(\mathrm{x}\right)\right)}^{{}^{1 - \mathrm{n}}}} + \mathrm{c}$

• $\frac{\mathrm{n} + 1}{{\left(1 + \ln \left(\mathrm{x}\right)\right)}^{{}^{ \mathrm{n} + 1}}} + \mathrm{c}$

• $\frac{1}{\left(\mathrm{n} - 1\right){\left(1 + \ln \left(\mathrm{x}\right)\right)}^{{}^{ \mathrm{n} - 1}}} + \mathrm{c}$

What is $\int \frac{\mathrm{dx}}{2{\mathrm{x}}^2 - 2\mathrm{x} \hspace{0.17em}+ 1} = ?$

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

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

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

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

What is the solution of the differential equation $\ln \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \mathrm{x}$

• y = e^x  + c

• y = e ^{- x}  + c

• y = ln(x) + c

• y = 2ln(x) + c

44.

What is the value of k such that integration of $\frac{3{\mathrm{x}}^2 + 8 - 4\mathrm{k}}{\mathrm{x}}$ with respect to x, may be a rational function ?

• 0

• 1

• 2

• - 2

What is $\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^{\mathrm{n}} \hspace{0.17em}1\right)} = ?$

• $\frac{1}{\mathrm{n}}\ln \left(\frac{{\mathrm{x}}^{\mathrm{n}}}{{\mathrm{x}}^{\mathrm{n}} + 1}\right) +\hspace{0.17em}\mathrm{c}$

• $\ln \left(\frac{{\mathrm{x}}^{\mathrm{n}} + 1}{{\mathrm{x}}^{\mathrm{n}}}\right) +\hspace{0.17em}\mathrm{c}$

• $\ln \left(\frac{{\mathrm{x}}^{\mathrm{n}}}{{\mathrm{x}}^{\mathrm{n}} + 1}\right) +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{\mathrm{n}}\ln \left(\frac{{\mathrm{x}}^{\mathrm{n}} + 1}{{\mathrm{x}}^{\mathrm{n}}}\right) +\hspace{0.17em}\mathrm{c}$

What is $\int \left({\mathrm{e}}^{\log \left(\mathrm{x}\right)} + \sin \left(\mathrm{x}\right)\right)\cos \left(\mathrm{x}\right)\mathrm{dx} = ?$

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

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

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

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

What is the value of ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left({\tan }^3\left(\mathrm{x}\right) + \tan \left(\mathrm{x}\right)\right)d\mathrm{x} ?$

• $\frac{1}{4}$$\frac{1}{4}$

• $\frac{1}{2}$

• 1

• 2

If $\mathrm{p}\left(\mathrm{x}\right) = {\left(4\mathrm{e}\right)}^{2\mathrm{x}}$, then what is $\int \mathrm{pdx} = ?$

• $\frac{\mathrm{p}\left(\mathrm{x}\right)}{1 + 2\ln \left(2\right)} + \mathrm{c}$

• $\frac{\mathrm{p}\left(\mathrm{x}\right)}{2\left(1 + 2\ln \left(2\right)\right)} + \mathrm{c}$

• $\frac{2\mathrm{p}}{1 + \ln \left(4\right)} + \mathrm{c}$

• $\frac{\mathrm{p}\left(\mathrm{x}\right)}{1 + \ln \left(2\right)} + \mathrm{c}$