First slide

Methods of integration

Question

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

Difficult

A

$π - 2$
B

$π + 2$
C

$2 (\sqrt{2} - 1)$
D

$2 \sqrt{2}$

Solution

$\int_{0}^{1} \sin^{- 1} (2 x \sqrt{1 - x^{2}}) d x, x = \sin θ$
= $\int_{0}^{\frac{π}{2}} \sin^{- 1} (\sin 2 θ) \cos θ d θ$

= $\int_{0}^{\frac{π}{4}} 2 θ \cos θ d θ + \int_{\frac{π}{4}}^{\frac{π}{2}} (π - 2 θ) \cos θ d θ$

$= 2 {[θ \sin θ + \cos θ]}_{0}^{\frac{π}{4}} + {[(π - 2 θ) \sin θ - 2 \cos θ]}_{\frac{π}{4}}^{\frac{π}{2}}$

$= \frac{π}{2 \sqrt{2}} + \frac{2}{\sqrt{2}} - 2 - \frac{π}{2 \sqrt{2}} + \frac{2}{\sqrt{2}} = 2 (\sqrt{2} - 1)$

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