First slide

Methods of integration

Question

$\int \frac{1}{x \sqrt{1 - x^{3}}} d x$ is equal to

Moderate

A

$\frac{1}{3} \log |\frac{\sqrt{1 - x^{3}} - 1}{\sqrt{1 - x^{3}} + 1}| + C$
B

$\frac{1}{3} \log |\frac{\sqrt{1 - x^{2}} + 1}{\sqrt{1 - x^{2}} - 1}| + C$
C

$\frac{1}{3} \log |\frac{1}{\sqrt{1 - x^{3}}}| + C$
D

none of these

Solution

We have,
$\begin{array}{l} l = \int \frac{1}{x \sqrt{1 - x^{3}}} d x \\ \Rightarrow I = - \frac{1}{3} \int \frac{1}{x^{3} \sqrt{1 - x^{3}}} (- 3 x^{2}) d x \\ \Rightarrow I = - \frac{1}{3} \int \frac{1}{x^{3} \sqrt{1 - x^{3}}} d (1 - x^{3}) \end{array}$
$\Rightarrow I = - \frac{1}{3} \int \frac{1}{(1 - t^{2}) \sqrt{t^{2}}} 2 t d t$ , where $t^{2} = 1 - x^{3}$
$\begin{array}{l} \Rightarrow t = - \frac{2}{3} \int \frac{1}{1 - t^{2}} d t = \frac{2}{3} \int \frac{1}{t^{2} - 1^{2}} d t = \frac{1}{3} \log |\frac{t - 1}{| t + 1 |}| + C \\ \Rightarrow I = \frac{1}{3} \log |\frac{\sqrt{1 - x^{3}} - 1}{\sqrt{1 - x^{3}} + 1}| + C \end{array}$

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