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Integrals

Multiple Choice Questions

821.

$\int_{0}^{\frac{π}{2}} \frac{\tan^{7} (x)}{\cot^{7} (x) + \tan^{7} (x)} dx$ is equal to

$\frac{π}{4}$
$\frac{π}{2}$
$\frac{π}{6}$
$\frac{π}{3}$

822.

$\int {xsec}^{2} (x) dx$ is equal to

$xtan (x) + \log (\sec (x)) + c$
$\frac{x^{2}}{2} \sec (x) + \log (\cos (x)) + c$
$xtan (x) + \log (\cos (x)) + c$
$\tan (x) + \log (\cos (x)) + c$

823.

$\int \frac{t}{e^{3 t^{2}}} dt$ is equal to

$\frac{1}{6} e^{3 t^{2}} + c$
$- \frac{1}{6} e^{3 t^{2}} + c$
$\frac{1}{6} e^{- 3 t^{2}} + c$
$- \frac{1}{6} e^{- 3 t^{2}} + c$

824.
$\int_{0}^{π} \log (\sin^{2} (x)) dx$ is equal to

$2 {πlog}_{e} (\frac{1}{2})$

${πlog}_{e} (2)$

$\frac{π}{2} \log_{e} (\frac{1}{2})$

None of these

$2 {πlog}_{e} (\frac{1}{2})$

$Let I = \int_{0}^{π} \log (\sin^{2} (x)) dx = 2 \int_{0}^{π} \log (\sin (x)) dx = 4 \int_{0}^{\frac{π}{2}} \log (\sin (x)) dx . . . (i) \Rightarrow I = 4 \int_{0}^{\frac{π}{2}} \log (\sin (\frac{π}{2} - x)) dx = 4 \int_{0}^{\frac{π}{2}} \log (\cos (x)) dx . . . (ii) On adding Eqs . (i) and (ii), we get 2 I = 4 \int_{0}^{\frac{π}{2}} \log (\sin (x) \cos (x)) dx \Rightarrow I = 2 \int_{0}^{\frac{π}{2}} \log [\sin (2 x) - \log (2)] dx = 2 \int_{0}^{\frac{π}{2}} \log (\sin (2 x)) dx - 2 {[xlog (2)]}_{0}^{\frac{π}{2}} = \frac{2}{2} \int_{0}^{π} \log (\sin (t)) - 2 . \frac{π}{2} \log_{e} (2) [∵ Putting 2 x = t]$

$= 2 \int_{0}^{\frac{π}{2}} \log (\sin (x) dx) - {πlog}_{e} (2) \Rightarrow I = 2 . \frac{I}{4} - {πlog}_{e} (2) [∵ using Eq . (i)] \Rightarrow I - \frac{I}{2} = - {πlog}_{e} (2) \Rightarrow I = - 2 {πlog}_{e} (2) = 2 {πlog}_{e} (\frac{1}{2})$