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Integrals

Multiple Choice Questions

741.

$\int \frac{dx}{x (x^{7} + 1)}$ is equal to

$\log (\frac{x^{7}}{x^{7} + 1}) + c$
$\frac{1}{7} \log (\frac{x^{7}}{x^{7} + 1}) + c$
$\log (\frac{x^{7} + 1}{x^{7}}) + c$
$\frac{1}{7} \log (\frac{x^{7} + 1}{x^{7}}) + c$

742.

$\int_{- 1}^{1} |1 - x| dx$ is equal to

743.

$\int \sqrt{x} e^{\sqrt{x}} dx$ is equal to

$2 \sqrt{x} - e^{\sqrt{x}} - 4 \sqrt{{xe}^{\sqrt{x}}} + c$
$(2 \sqrt{x} - 4 \sqrt{x} + 4) e^{\sqrt{x}} + c$
$(2 \sqrt{x} + 4 \sqrt{x} + 4) e^{\sqrt{x}} + c$
$(1 - 4 \sqrt{x}) e^{\sqrt{x}} + c$

744.

$\int \frac{dx}{x^{2} + 2 x + 2}$ is equal to

$\sin^{- 1} (x + 1) + c$
$\sinh^{- 1} (x + 1) + c$
$\tanh^{- 1} (x + 1) + c$
$\tan^{- 1} (x + 1) + c$

745.

$\int_{0}^{2 π} (\sin (x) + |\sin (x)|) dx$ is equal to

746.

$\int_{0}^{π} \frac{xdx}{a^{2} \cos^{2} (x) + b^{2} \sin^{2} (x)} dx$ is equal to

$\frac{π}{2 ab}$
$\frac{π}{ab}$
$\frac{π^{2}}{2 ab}$
$\frac{π^{2}}{ab}$

747.

$\int e^{x} (\frac{1 + \sin (x)}{1 + \cos (x)}) dx$ is equal to

$e^{x} \sec^{2} (\frac{x}{2}) + c$
$e^{x} \tan (\frac{x}{2})$ + c
$e^{x} \sec (\frac{x}{2}) + c$
$e^{x} \tan (x) + c$

748.
$\int \sqrt{1 + \sin (\frac{x}{4})} dx$ is equal to

$8 (\sin (\frac{x}{8}) + \cos (\frac{x}{8})) + C$

$8 (\sin (\frac{x}{8}) - \cos (\frac{x}{8})) + C$

$8 (\cos (\frac{x}{8}) - \sin (\frac{x}{8})) + C$

$\frac{1}{8} (\sin (\frac{x}{8}) - \cos (\frac{x}{8})) + C$

$8 (\sin (\frac{x}{8}) - \cos (\frac{x}{8})) + C$

$We have, \int \sqrt{1 + \sin (\frac{x}{4})} dx = \int \sqrt{(1 + 2 \sin (\frac{x}{8}) \cos (\frac{x}{8}))} dx [∵ \sin (2 x) = 2 \sin (x) \cos (x)] = \int \sqrt{(\sin^{2} (\frac{x}{8}) + \cos^{2} (\frac{x}{8})) + 2 \sin (\frac{x}{8}) \cos (\frac{x}{8})} dx = \sqrt{{(\sin (\frac{x}{8}) - \cos (\frac{x}{8}))}^{2}} dx [∵ {(a + b)}^{2} = a^{2} + b^{2} + 2 ab] = \int (\sin (\frac{x}{8}) + \cos (\frac{x}{8})) dx = \frac{- \cos (\frac{x}{8})}{\frac{1}{8}} + \frac{\sin (\frac{x}{8})}{\frac{1}{8}} = 8 (- \cos (\frac{x}{8}) + \sin (\frac{x}{8})) + C = 8 (- \cos (\frac{x}{8}) + \sin (\frac{x}{8})) = 8 (\sin (\frac{x}{8}) - \cos (\frac{x}{8})) + C$