${\int }_0^1\frac{\mathrm{dx}}{\sqrt{1 + \mathrm{x}} + \sqrt{\mathrm{x}}}$ is equal to

• $\frac{4}{3}\left(\sqrt{2} - 1\right)$

• $\frac{3}{4}\left(\sqrt{2} - 1\right)$

• $\frac{4}{3}\left(1 - \sqrt{2}\right)$

• $\frac{3}{4}\left(1 - \sqrt{2}\right)$

${\int }_0^1\left|5\mathrm{x} - 3\right|\mathrm{dx}$ is equal to

• $\frac{10}{13}$

• $\frac{31}{10}$

• $\frac{13}{10}$

• None of these

${\int }_0^{\mathrm{\pi }}{\mathrm{xsin}}^4\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{3\mathrm{\pi }}{16}$

• $\frac{3{\mathrm{\pi }}^2}{16}$

• $\frac{16\mathrm{\pi }}{3}$

• $\frac{16{\mathrm{\pi }}^2}{3}$

714.

$\int 5^{5^{5^{\mathrm{x}}}} 5^{5^{\mathrm{x}}} 5^{\mathrm{x}} \mathrm{dx}$ is equal to

• $5^{5^{5^{\mathrm{x}}}}{\left(\log \left(5\right)\right)}^3 + \mathrm{c}$

• $\frac{5^{5^{5^{\mathrm{x}}}}}{{\left(\log \left(5\right)\right)}^3} + \mathrm{c}$

• $\frac{5^{{}^{5^{\mathrm{x}}}}}{{\left(\log \left(5\right)\right)}^3} + \mathrm{c}$

• $5^{{}^{5^{\mathrm{x}}}}{\left(\log \left(5\right)\right)}^3 + \mathrm{c}$

The value of ${\int }_2^4\left\{\left|\mathrm{x} - 2\right| + \left|\mathrm{x} - 3\right|\right\}\mathrm{dx}$ is

• 1 sq unit

• 2 sq unit

• 3 sq unit

• 5 sq unit

If b > a, then ${\int }_{\mathrm{a}}^{\mathrm{b}}\frac{\mathrm{dx}}{\sqrt{\left(\mathrm{x} - \mathrm{a}\right)\left(\mathrm{b} - \mathrm{x}\right)}}$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}\left(\mathrm{b} - \mathrm{a}\right)$

• $\frac{\mathrm{\pi }}{4}\left(\mathrm{b} - \mathrm{a}\right)$

$\int {\mathrm{e}}^{\mathrm{x}}\frac{{\mathrm{x}}^2 + 1}{{\left(\mathrm{x} + 1\right)}^2}\mathrm{dx}$ is equal to

• $\frac{- {\mathrm{e}}^{\mathrm{x}}}{\mathrm{x} +\hspace{0.17em}1} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x} +\hspace{0.17em}1} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}\left(\mathrm{x} - 1\right)}{\mathrm{x} +\hspace{0.17em}1} + \mathrm{c}$

• $\frac{{\mathrm{xe}}^{\mathrm{x}}}{\mathrm{x} +\hspace{0.17em}1} + \mathrm{c}$

$\int {\csc }^4\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\cot \left(\mathrm{x}\right) + \frac{{\cot }^3\left(\mathrm{x}\right)}{3} +\hspace{0.17em}\mathrm{C}$

• $\tan \left(\mathrm{x}\right) + \frac{{\tan }^3\left(\mathrm{x}\right)}{3} +\hspace{0.17em}\mathrm{C}$

• $- \cot \left(\mathrm{x}\right) - \frac{{\cot }^3\left(\mathrm{x}\right)}{3} +\hspace{0.17em}\mathrm{C}$

• $- \tan \left(\mathrm{x}\right) - \frac{{\tan }^3\left(\mathrm{x}\right)}{3} +\hspace{0.17em}\mathrm{C}$

${\int }_0^{1000}{\mathrm{e}}^{\mathrm{x} - \left[\mathrm{x}\right]}\mathrm{dx}$ is

• e¹⁰⁰⁰ - 1

• $\frac{{\mathrm{e}}^{1000} - 1}{\mathrm{e} - 1}$

• 1000(e - 1)

• $\frac{\mathrm{e} - 1}{1000}$

$\int \frac{1 + \mathrm{x} + \sqrt{\mathrm{x} +\hspace{0.17em}{\mathrm{x}}^2}}{\sqrt{\mathrm{x}} + \sqrt{1 + \mathrm{x}}}\mathrm{dx}$ is equal to

• $\frac{1}{2}\sqrt{1 + \mathrm{x}} + \mathrm{C}$

• $\frac{2}{3}{\left(1 + \mathrm{x}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + \mathrm{C}$

• $\sqrt{1 + \mathrm{x}} +\hspace{0.17em}\mathrm{C}$

• 2(1 + x)^3/2 + C