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Integrals

Multiple Choice Questions

451.

$\int_{0}^{π} \frac{xdx}{1 + \sin (x)}$ is equal to :

$- π$
$\frac{π}{2}$
$π$
None of these

452.

$\int \frac{dx}{x (x^{5} + 1)}$ is equal to :

$\frac{1}{5} \log (x^{5}) (x^{5} + 1) + c$
$\frac{1}{5} \log (\frac{x^{5} + 1}{x^{5}}) + c$
$\frac{1}{5} \log (\frac{x^{5}}{x^{5} + 1}) + c$
None of these

453.

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

$xtan (\frac{x}{2}) + c$
${xsec}^{2} (\frac{x}{2}) + c$
$\log (\cos (\frac{x}{2})) + c$
None of these

454.

$\int_{0}^{\frac{π}{8}} \cos^{3} (4 θ) dθ$ is equal to

$\frac{5}{3}$
$\frac{5}{4}$
$\frac{1}{3}$
$\frac{1}{6}$

455.

$\int_{0}^{\frac{π}{2}} \frac{\cos (x) - \sin (x)}{1 + \cos (x) \sin (x)} dx$ is equal to

0
$\frac{π}{2}$
$\frac{π}{4}$
$\frac{π}{6}$

456.
$\int \frac{dx}{x (x^{7} + 1)}$ is equal to

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

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

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

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

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

$Given that, \int \frac{dx}{x (x^{7} + 1)} On putting x^{7} = t \Rightarrow 7 x^{6} dx = dt \Rightarrow dx = \frac{1}{7 x^{6}} dt ∴ I = \int \frac{1}{7 x^{7}} \frac{dt}{(t + 1)} = \frac{1}{7} \int \frac{1}{t} \frac{dt}{(t + 1)} [∵ x^{7} = t] = \frac{1}{7} \int (\frac{1}{t} - \frac{1}{t + 1}) = \frac{1}{7} \log (\frac{t}{t + 1}) + c = \frac{1}{7} \log (\frac{x^{7}}{x^{7} + 1}) + c$