If f(x) = $\cos \left(\mathrm{x}\right) - {\cos }^2\left(\mathrm{x}\right) + {\cos }^3\left(\mathrm{x}\right) - ... \mathrm{to} \infty$, then $\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ equals to

• $\tan \left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• $\mathrm{x} + \tan \left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

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

• $\mathrm{x} - \tan \left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

${\int }_0^1\frac{\mathrm{xdx}}{\left[\mathrm{x} + \sqrt{1 - {\mathrm{x}}^2}\right]\sqrt{1 - {\mathrm{x}}^2}}$ is equal to :

• 0

• 1

• $\frac{\mathrm{\pi }}{4}$

• $\frac{{\mathrm{\pi }}^2}{2}$

283.

The value of ${\int }_0^{2\mathrm{a}}\frac{\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}}{\mathrm{f}\left(\mathrm{x}\right) + \mathrm{f}\left(2\mathrm{a} - \mathrm{x}\right)}$ is :

• f(a)

• f(2a)

• f(0)

• a

The value of ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\mathrm{x}}^3{\sin }^4\left(\mathrm{x}\right)\mathrm{dx}$ is equal to :

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{8}$

• 0

For any positive integer n, $\int \frac{\mathrm{dx}}{{\mathrm{x}}^{\mathrm{n} + 1} + \mathrm{x}}$ is equal to :

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

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

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

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

$\int \cos \left[2{\cot }^{-1}\left(\sqrt{\frac{1 - \mathrm{x}}{1 + \mathrm{x}}}\right)\right]\mathrm{dx}$ is equal to :

• $\frac{1}{2}{\mathrm{x}}^2 + \mathrm{c}$

• $\frac{1}{2}\sin \left[2{\cot }^{-1}\left(\sqrt{\frac{1 - \mathrm{x}}{1 + \mathrm{x}}}\right)\right] + \mathrm{c}$

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

• $\frac{1}{2}\mathrm{x} + \mathrm{c}$

If the tangent to the graph function y = f(x) makes angles $\frac{\mathrm{\pi }}{4} \mathrm{and} \frac{\mathrm{\pi }}{3}$ with the x-axis is at the point x = 2 and x = 4 respectively, the value of ${\int }_2^4\mathrm{f}\text{'}\left(\mathrm{x}\right)\mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)\mathrm{dx}$ :

• f(4) f(2)

• f(4)

• f(2)

• 1

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\cos \left(\mathrm{x}\right)}{1 + \sin \left(\mathrm{x}\right)}\mathrm{dx}$ equals to :

• $\log \left(2\right)$

• $2\log \left(2\right)$

• ${\left(\log \left(2\right)\right)}^2$

• $\frac{1}{2}\log \left(2\right)$

$\int \frac{\mathrm{dx}}{\sin \left(\mathrm{x} - \mathrm{a}\right)\sin \left(\mathrm{x} - \mathrm{b}\right)}$ is :

• $\frac{1}{\sin \left(\mathrm{a} - \mathrm{b}\right)}\log \left|\frac{\sin \left(\mathrm{x} - \mathrm{a}\right)}{\sin \left(\mathrm{x} - \mathrm{b}\right)}\right|$ + c

• $\frac{- 1}{\sin \left(\mathrm{a} - \mathrm{b}\right)}\log \left|\frac{\sin \left(\mathrm{x} - \mathrm{a}\right)}{\sin \left(\mathrm{x} - \mathrm{b}\right)}\right|$ + c

• $\log \left(\sin \left(\mathrm{x} - \mathrm{a}\right)\sin \left(\mathrm{x} - \mathrm{b}\right)\right)$ + c

• $\log \left|\frac{\sin \left(\mathrm{x} - \mathrm{a}\right)}{\sin \left(\mathrm{x} - \mathrm{b}\right)}\right|$

If an antiderivative of f(x) is e^x and that of g(x) is cos(x), then $\int \mathrm{f}\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)\mathrm{dx}$ + $\int \mathrm{g}\left(\mathrm{x}\right){\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ is equal to :

• f(x)g(x) + c

• f(x) + g(x) + c

• e^xcos(x) + c

• f(x) - g(x) + c