$\int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}\right) + \sqrt{3}\cos \left(\mathrm{x}\right)}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{2} - \frac{\mathrm{\pi }}{6}\right)\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{4} - \frac{\mathrm{\pi }}{6}\right)\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{6}\right)\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{4} + \frac{\mathrm{\pi }}{3}\right)\right| + \mathrm{c}$

If f(x) = f(a - x), then ${\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is equal to

• ${\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $\frac{{\mathrm{a}}^2}{2}{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $\frac{\mathrm{a}}{2}{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $- \frac{\mathrm{a}}{2}{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

The value of ${\int }_0^{\infty }\frac{d\mathrm{x}}{\left({\mathrm{x}}^2 + 4\right)\left({\mathrm{x}}^2 + 9\right)}$ is

• $\frac{\mathrm{\pi }}{60}$

• $\frac{\mathrm{\pi }}{20}$

• $\frac{\mathrm{\pi }}{40}$

• $\frac{\mathrm{\pi }}{80}$

If I₁ = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\sin }^2\left(\mathrm{x}\right)d\mathrm{x}$ and I₂ = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\cos }^2\left(\mathrm{x}\right)d\mathrm{x}$, then

• I₁ = I₂

• I₁ < I₂

• I₁ > I₂

• I₂ = I₁ + $\frac{\mathrm{\pi }}{4}$

The integrating factor of the differential equation $\mathrm{xlog}\left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = 2\log \left(\mathrm{x}\right)$ is given by

• e^x

• log(x)

• log(log(x))

• x

If ${\int }_{- 1}^4\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = 4 \mathrm{and} {\int }_2^4\left\{3 - \mathrm{f}\left(\mathrm{x}\right)\right\}d\mathrm{x} = 7$, then the value of

${\int }_{- 1}^2\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is

• - 2

• 3

• 4

• 5

If m, n be integers, then find the value of

${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}{\left(\cos \left(\mathrm{mx}\right) - \sin \left(\mathrm{nx}\right)\right)}^{2}d\mathrm{x}$

If I = ${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}\frac{{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}{{\mathrm{e}}^{\sin \left(\mathrm{x}\right)} +\hspace{0.17em}{\mathrm{e}}^{- \sin \left(\mathrm{x}\right)}}d\mathrm{x}$, then I equals

• $\frac{\mathrm{\pi }}{2}$

• $2\mathrm{\pi }$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

If h(x) = ${\int }_0^{\mathrm{x}}{\sin }^4\left(\mathrm{t}\right)d\mathrm{t}, \mathrm{then} \mathrm{h}\left(\mathrm{x} + \mathrm{\pi }\right)$ equals

• $\frac{\mathrm{h}\left(\mathrm{x}\right)}{\mathrm{h}\left(\mathrm{\pi }\right)}$

• $\mathrm{h}\left(\mathrm{x}\right)\mathrm{h}\left(\mathrm{\pi }\right)$

• $\mathrm{h}\left(\mathrm{x}\right) - \mathrm{h}\left(\mathrm{\pi }\right)$

• $\mathrm{h}\left(\mathrm{x}\right) + \mathrm{h}\left(\mathrm{\pi }\right)$

The value of the integral ${\int }_0^2\left|{\mathrm{x}}^2 - 1\right|d\mathrm{x}$ is

• 0

• 2

• - $\frac{1}{3}$

• - 2