First slide

Methods of integration

Question

$\begin{array}{l} If \int \frac{1}{x \sqrt{1 - x^{3}}} d x = a \log |\frac{\sqrt{1 - x^{3}} - 1}{\sqrt{1 - x^{3}} + 1}| + b, then a is equal to \end{array}$

Easy

A

$\frac{1}{3}$
B

$\frac{- 1}{3}$
C

$\frac{2}{3}$
D

$\frac{- 2}{3}$

Solution

$\begin{array}{l} L e t 1 - x^{3} = y^{2} . T h e n - 3 x^{2} d x = 2 y d y \\ ∴ \int \frac{1}{x \sqrt{1 - x^{3}}} d x = \int \frac{x^{2} d x}{x^{3} \sqrt{1 - x^{3}}} \\ = - \frac{2}{3} \int \frac{y d y}{(1 - y^{2}) y} \\ = \frac{1}{3} \log |\frac{y - 1}{y + 1}| + C \\ = \frac{1}{3} \log |\frac{\sqrt{1 - x^{3}} - 1}{\sqrt{1 - x^{3}} + 1}| + C \\ ∴ a = \frac{1}{3} \end{array}$

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