First slide

Evaluation of definite integrals

Question

$\int_{0}^{1 / \sqrt{2}} \frac{\sin^{- 1} x}{{(1 - x^{2})}^{3 / 2}} d x =$

Easy

A

$\frac{π}{4} + \log 2$
B

$\frac{π}{4} - \log 2$
C

$\frac{π}{4} - \frac{1}{2} \log 2$
D

$\log 2$

Solution

$\int_{0}^{1 / \sqrt{2}} \frac{S i n^{- 1} x}{{(1 - x^{2})}^{3 / 2}} d x$          $t = \sin^{- 1} x$
$= \int_{0}^{π / 4} \frac{t}{C o s^{2} t} d t = \int_{0}^{π / 4} t S e c^{2} t d t$

                          $= {(t T a n t)}_{0}^{π / 4} - \int_{0}^{π / 4} T a n t d t$

                           $= \frac{π}{4} - {(\log 1 \sqrt{C o t 1})}_{0}^{π / 4}$

                         $= \frac{π}{4} - \log \sqrt{2}$

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