Mathematics : Integrals

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \frac{1}{{\mathrm{x}}^2}{\int }_3^{\mathrm{x}}\left(2\mathrm{t} - 3\mathrm{f}\text{'}\left(\mathrm{t}\right)\right)\mathrm{dt}, \mathrm{then} \mathrm{f}\text{'}\left(\mathrm{t}\right), \mathrm{then} \mathrm{f}\text{'}\left(3\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{- 1 }{2}$

• $\frac{- 1}{3}$

• $\frac{1}{2}$

• $\frac{1}{3}$

$\int \frac{\mathrm{dx}}{\left(\mathrm{x} + 100\right)\sqrt{\mathrm{x} + 99}} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c} \Rightarrow \mathrm{f}\left(\mathrm{x}\right)$

• 2(x + 100)^1/2

• 3(x + 100)^1/2

• $2{\tan }^{-1}\left(\sqrt{\mathrm{x} + 99}\right)$

• $2{\tan }^{-1}\left(\sqrt{\mathrm{x} + 100}\right)$

$\int \frac{3 - {\mathrm{x}}^2}{1 - 2\mathrm{x} + {\mathrm{x}}^2}{\mathrm{e}}^{\mathrm{x}}\mathrm{dx} = {\mathrm{e}}^{\mathrm{x}}\mathrm{f}\left(\mathrm{x}\right) + \mathrm{c} \Rightarrow \mathrm{f}\left(\mathrm{x}\right)$

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

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

• $\frac{1 - \mathrm{x}}{\mathrm{x} - 1}$

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

$\int \frac{\sqrt{\cot \left(\mathrm{x}\right)}}{\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}\mathrm{dx} = - \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c} \Rightarrow \mathrm{f}\left(\mathrm{x}\right)$

• $2\sqrt{\tan \left(\mathrm{x}\right)}$

• $- 2\sqrt{\tan \left(\mathrm{x}\right)}$

• $- 2\sqrt{\cot \left(\mathrm{x}\right)}$

• $2\sqrt{\cot \left(\mathrm{x}\right)}$

${\int }_{\frac{- \mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\log \left(\frac{2 - \sin \left(\mathrm{\theta }\right)}{2 + \sin \left(\mathrm{\theta }\right)}\right)\mathrm{d\theta }$ is equal to

• 0

• 1

• 2

• - 1

${\int }_0^2\frac{2\mathrm{x} - 2}{2\mathrm{x} - {\mathrm{x}}^2}\mathrm{dx}$ is equal to

• 0

• 2

• 3

• 4

If $\int \frac{\sin \left(\mathrm{x}\right)}{\cos \left(\mathrm{x}\right)\left(1 + \cos \left(\mathrm{x}\right)\right)}\mathrm{dx}$ = f(x) + c, then f(x) is equal to

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

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

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

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

$\int \frac{{\mathrm{x}}^{49}{\tan }^{-1}\left({\mathrm{x}}^{50}\right)}{\left(1 + {\mathrm{x}}^{100}\right)}\mathrm{dx} = \mathrm{k}{\left({\tan }^{-1}\left({\mathrm{x}}^{50}\right)\right)}^2 +\hspace{0.17em}\mathrm{c}$, then k is equal to

• $\frac{1}{50}$

• $- \frac{1}{50}$

• $\frac{1}{100}$

• $- \frac{1}{100}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{200\sin \left(\mathrm{x}\right) + 100\cos \left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $50\mathrm{\pi }$

• $25\mathrm{\pi }$

• $75\mathrm{\pi }$

• $150\mathrm{\pi }$

${\int }_0^{\mathrm{\pi }}\frac{\mathrm{\theta sin}\left(\mathrm{\theta }\right)}{1 + {\cos }^2\left(\mathrm{\theta }\right)}\mathrm{d\theta }$ is equal to

• $\frac{{\mathrm{\pi }}^2}{2}$

• $\frac{{\mathrm{\pi }}^2}{3}$

• ${\mathrm{\pi }}^2$

• $\frac{{\mathrm{\pi }}^2}{4}$