Mathematics : Integrals

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

$\int {\mathrm{e}}^{\mathrm{xlog}\left(\mathrm{a}\right)}{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ is equal to :

• $\frac{{\mathrm{a}}^{\mathrm{x}}}{\log \left(\mathrm{ae}\right)} + \mathrm{c}$

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

• ${\left(\mathrm{ae}\right)}^{\mathrm{x}} + \mathrm{c}$

• $\frac{{\displaystyle {\left(\mathrm{ae}\right)}^{\mathrm{x}}}}{{\displaystyle {\log }_e\left(\mathrm{ae}\right)}} + \mathrm{c}$

$\int \sqrt{{\mathrm{e}}^{\mathrm{x}} - 1}\mathrm{dx}$ is equal to :

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

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

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

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

If I₁ = $\int {\sin }^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ and I₂ = $\int {\sin }^{-1}\left(\sqrt{1 - {\mathrm{x}}^2}\right)\mathrm{dx}$ then : 

• I₁ = I₂

• ${\mathrm{I}}_{2 }= \frac{\mathrm{\pi }}{2}{\mathrm{I}}_1$

• I₁ + I₂ = $\frac{\mathrm{\pi }}{2}\mathrm{x}$

• I₁ + I₂ = $\frac{\mathrm{\pi }}{2}$

$\int {\cos }^{\raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right){\sin }^{\raisebox{1ex}{$-11$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to:

• $\log \left|{\sin }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right)\right| + \mathrm{c}$

• $\frac{4}{7}{\tan }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right) + \mathrm{c}$

• $\frac{- 7}{4}{\tan }^{\raisebox{1ex}{$-4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right) + \mathrm{c}$

• $\log \left|{\cos }^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right)\right| + \mathrm{c}$

$\int \frac{\left(\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)\right)}{\sqrt{\sin \left(2\mathrm{\theta }\right)}}\mathrm{d\theta }$ is equal to :

• $\log \left|\cos \left(\mathrm{\theta }\right) - \sin \left(\mathrm{\theta }\right) + \sqrt{\sin \left(2\mathrm{\theta }\right)}\right|$

• $\log \left|\sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right) + \sqrt{\sin \left(2\mathrm{\theta }\right)}\right|$

• ${\sin }^{-1}\left(\sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right)\right)$

• ${\sin }^{-1}\left(\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)\right)$