21.

What is the solution of the differential equation

$\log \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) - \mathrm{a} = 0$

• y = xe^a + c

• x = ye^a + c

• $\mathrm{y} = \ln \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{x} = \ln \left(\mathrm{y}\right) + \mathrm{c}$

${\int }_{{\mathrm{e}}^{ - 1}}^{{\mathrm{e}}^{ - 2}}\left|\frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}\right|d\mathrm{x} = ?$

• $\frac{3}{2}$

• $\frac{5}{2}$

• 3

• 4

What is $\int {\tan }^{-1}\left(\sec \left(\mathrm{x}\right) + \tan \left(\mathrm{x}\right)\right)\mathrm{dx} = ?$

• $\frac{\mathrm{\pi x}}{4} +\hspace{0.17em}\frac{{\mathrm{x}}^2}{4} +\hspace{0.17em}\mathrm{c}$

• $\frac{\mathrm{\pi x}}{2} + \frac{{\mathrm{x}}^2}{4} + \mathrm{c}$

• $\frac{\mathrm{\pi x}}{4} +\hspace{0.17em}\frac{{\mathrm{\pi x}}^2}{4} + \mathrm{c}$

• $\frac{\mathrm{\pi x}}{4} - \frac{{\mathrm{x}}^2}{4} + \mathrm{c}$

What is ${\int }_0^{2\mathrm{\pi }}\sqrt{1 + \sin \left(\frac{\mathrm{x}}{2}\right)}d\mathrm{x} = ?$

• 8

• 4

• 2

• 0

$\int \left(\ln {\left(\mathrm{x}\right)}^{ - 1}\right)\mathrm{dx} - \int \left(\ln {\left(\mathrm{x}\right)}^{ - 2}\right)\mathrm{dx} = ?$

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

• $\mathrm{x}{\left(\ln \left(\mathrm{x}\right)\right)}^{ - 2}$ = c

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

• $\mathrm{x}{\left(\ln \left(\mathrm{x}\right)\right)}^2 + \mathrm{c}$

The value of ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{\tan \left(\mathrm{x}\right)}d\mathrm{x} + {\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{\cot \left(\mathrm{x}\right)}d\mathrm{x} = ?$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2\sqrt{2}}$

• $\frac{\mathrm{\pi }}{\sqrt{2}}$

Consider the functions

f(x) = xg(x) and g(x) = $\left[\frac{1}{\mathrm{x}}\right]$

where [ . ] is the greatest integer function

• $\frac{1}{6}$

• $\frac{1}{3}$

• $\frac{5}{18}$

• $\frac{5}{36}$

Consider the functions

f(x) = xg(x) and g(x) = $\left[\frac{1}{\mathrm{x}}\right]$

where [ . ] is the greatest integer function

What is ${\int }_{\frac{1}{3}}^1\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $\frac{37}{72}$

• $\frac{2}{3}$

• $\frac{17}{72}$

• $\frac{37}{144}$

Given that

${\mathrm{a}}_{\mathrm{n}} = {\int }_0^{\mathrm{\pi }}\frac{\sin \left\{\left(\mathrm{n} + 1\right)\mathrm{x}\right\}}{\sin \left(2\mathrm{x}\right)}d\mathrm{x}$

Consider the following statements :

1. The sequence {${\mathrm{a}}_{2\mathrm{n}}$ } is in AP with common difference zero.
2. The sequence {${\mathrm{a}}_{2\mathrm{n} + 1}$} is in AP with common difference zero.

Which of the above statements is/are correct ?

• 1 only

• 2 only

• both 1 and 2

• neither 1 nor 2

Given that

${\mathrm{a}}_{\mathrm{n}} = {\int }_0^{\mathrm{\pi }}\frac{\sin \left\{\left(\mathrm{n} + 1\right)\mathrm{x}\right\}}{\sin \left(2\mathrm{x}\right)}d\mathrm{x}$

What is a_{n - 1} - a_{n - 4} = ?

•  - 1

• 0

• 1

• 2