First slide

Evaluation of definite integrals

Question

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

Easy

A

$\frac{π}{2}$
B

$\frac{π}{2} + \ln 2$
C

$\frac{π}{2} - \ln 2$
D

ln 2

Solution

$\int_{0}^{1} S i n^{- 1} (\frac{2 x}{1 + x^{2}}) d x = \int_{0}^{1} 2 T a n^{- 1} x d x$
                                   $= 2 {[T a n^{- 1} x . x]}_{0}^{1} - 2 \int_{0}^{1} \frac{1}{1 + x^{2}} x d x$

                                   $= 2 (\frac{π}{4} - 0) - {[\log (1 + x^{2})]}_{0}^{1}$

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

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