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

• 0

• 2

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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

• $\frac{{\mathrm{\pi }}^2}{2}$

• $\frac{{\mathrm{\pi }}^2}{3}$

• ${\mathrm{\pi }}^2$

• $\frac{{\mathrm{\pi }}^2}{4}$

If $\int {\sin }^{-1}\left(\frac{2\mathrm{x}}{1 + {\mathrm{x}}^2}\right)\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) - \log \left(1 + {\mathrm{x}}^2\right)$ then f(x) is equal to

• 2xtan^-1(x)

• - 2xtan^-1(x)

• xtan^-1(x)

• - xtan^-1(x)

$\mathrm{If} \int \sqrt{\frac{\mathrm{x}}{{\mathrm{a}}^3 - {\mathrm{x}}^3}}\mathrm{dx} = \mathrm{g}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}, \mathrm{then} \mathrm{g}\left(\mathrm{x}\right) \mathrm{is} \mathrm{equal} \mathrm{to} :$

• $\frac{2}{3}{\cos }^{-1}\left(\mathrm{x}\right)$

• $\frac{2}{3}{\sin }^{-1}\left(\frac{{\mathrm{x}}^3}{{\mathrm{a}}^3}\right)$

• $\frac{2}{3}{\sin }^{-1}\left(\sqrt{\frac{{\mathrm{x}}^3}{{\mathrm{a}}^3}}\right)$

• $\frac{2}{3}{\cos }^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)$

If $\int \frac{\mathrm{dx}}{{\mathrm{x}}^2 + 2\mathrm{x} + 2} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c}$, then f(x) is equal to

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

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

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

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

Observe the following statement:

$$\begin{gathered} \mathrm{A} : \int \left(\frac{{\mathrm{x}}^2 - 1}{{\mathrm{x}}^2}\right){\mathrm{e}}^{\frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2}}\mathrm{dx} = {\mathrm{e}}^{\frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2}} + \mathrm{c} \\ \mathrm{R} : \int \mathrm{f}\text{'}\left(\mathrm{x}\right){\mathrm{e}}^{\mathrm{f}\left(\mathrm{x}\right)}\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c} \end{gathered}$$

Then which of the followmg is true ?

• Both A and R are true and R is not thecorrect reason for A

• Both A and R are true and R is the correct reason for A

• A is true, R is false

• A is false, R is true

Dividing the interval [0, 6] into 6 equal parts and by using trapezoidal rule the value of ${\int }_0^6{\mathrm{x}}^3\mathrm{dx}$ is approximately

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