The value of $\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^{\mathrm{n}} + 1\right)}$ is

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

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

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

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

The value of $\int \cos \left(\log \left(\mathrm{x}\right)\right)\mathrm{dx}$ is

• $\frac{1}{2}\left\{\sin \left(\log \left(\mathrm{x}\right)\right) + \cos \left(\log \left(\mathrm{x}\right)\right)\right\} + \mathrm{C}$

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

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

• $\frac{1}{2}\left\{\sin \left(\log \left(\mathrm{x}\right)\right) - \cos \left(\log \left(\mathrm{x}\right)\right)\right\} + \mathrm{C}$

The value of $\int {\mathrm{e}}^{\mathrm{x}}\left[\frac{1 + \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\right]\mathrm{dx}$ is

• $\frac{1}{2}{\mathrm{e}}^{\mathrm{x}}\sec \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

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

• $\frac{1}{2}{\mathrm{e}}^{\mathrm{x}}\tan \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

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

The value of $\int \frac{1}{3\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) + 3}\mathrm{dx}$ is

• $\log \left(\frac{\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}{2\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}\right)$ + C

• $\frac{1}{2}\log \left(\frac{2\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}{\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}\right)$ + C

• $\log \left(\frac{2\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}{\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}\right)$ + C

• $2\log \left(\frac{2\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}{\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}\right)$ + C

The value of $\int \frac{\sin \left(2\mathrm{x}\right)}{{\sin }^4\left(\mathrm{x}\right) + {\cos }^4\left(\mathrm{x}\right)}\mathrm{dx}$ is

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

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

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

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

The value of $\int \sqrt{1 + \sec \left(\mathrm{x}\right)}\mathrm{dx}$ is

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

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

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

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

The value of $\int \frac{\left({\mathrm{x}}^2 + 1\right)}{{\mathrm{x}}^4 +\hspace{0.17em}{\mathrm{x}}^2 + 1}\mathrm{dx}$ is

• $\frac{1}{\sqrt{3}}{\tan }^{-1}\left\{\frac{\mathrm{x} - 1/\mathrm{x}}{\sqrt{3}}\right\} + \mathrm{C}$

• $\frac{1}{2\sqrt{3}}\log \left\{\frac{\left(\mathrm{x} - 1/\mathrm{x}\right) - \sqrt{3}}{\left(\mathrm{x} - 1/\mathrm{x}\right) + \sqrt{3}}\right\} + \mathrm{C}$

• ${\tan }^{-1}\left\{\frac{\mathrm{x} + 1/\mathrm{x}}{\sqrt{3}}\right\} + \mathrm{C}$

• ${\tan }^{-1}\left\{\frac{\mathrm{x} - 1/\mathrm{x}}{\sqrt{3}}\right\} + \mathrm{C}$

The value of ${\int }_0^1{\mathrm{x}}^2{\left(1 - {\mathrm{x}}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{dx}$ is

• $\frac{1}{32}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{32}$

The value of ${\int }_0^{\infty }\frac{\mathrm{x}}{\left(1 + \mathrm{x}\right)\left({\mathrm{x}}^2 + 1\right)}\mathrm{dx}$ is

• $2\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{32}$

390.

$\int \frac{1}{\sqrt{8 + 2\mathrm{x} - {\mathrm{x}}^2}}\mathrm{dx}$ is equal to

• $\frac{1}{3}{\sin }^{-1}\left(\frac{\mathrm{x} - 1}{3}\right) + \mathrm{c}$

• ${\sin }^{-1}\left(\frac{\mathrm{x} + 1}{3}\right) + \mathrm{c}$

• $\frac{1}{3}{\sin }^{-1}\left(\frac{\mathrm{x} + 1}{3}\right) + \mathrm{c}$

• ${\sin }^{-1}\left(\frac{\mathrm{x} - 1}{3}\right) + \mathrm{c}$