First slide

Evaluation of definite integrals

Question

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

Difficult

A

$\frac{1}{2} + \frac{π}{2 \sqrt{3}}$
B

$\frac{1}{2} - \frac{π}{2 \sqrt{3}}$
C

$\frac{1}{2} + \frac{π}{4 \sqrt{3}}$
D

$\frac{1}{2} - \frac{π}{4 \sqrt{3}}$

Solution

$\int_{0}^{\frac{1}{2}} \frac{x \sin^{- 1}}{\sqrt{1 - x^{2}}} d x, x = \sin θ$
${= \int_{0}^{\frac{π}{6}} θ \sin θ d θ = - θ \cos θ + \sin θ |}_{0}^{\frac{π}{6}}$

$= - \frac{π}{6} \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{1}{2} - \frac{π}{4 \sqrt{3}}$

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