First slide

Introduction to integration

Question

The value of integral $\int \frac{d x}{x \sqrt{1 - x^{3}}}$ is given equal to

Easy

A

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

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

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

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

Solution

Put $1 - x^{3} = t^{2} .$ Then $- 3 x^{2} d x = 2 t d t$ and the entegral becomes
$\begin{array}{l} - \frac{1}{3} \int \frac{- 3 x^{2} d x}{x^{3} \sqrt{1 - x^{3}}} = - \frac{1}{3} \int \frac{2 t d t}{(1 - t^{2}) t} = \frac{2}{3} \int \frac{d t}{t^{2} - 1} \\ = \frac{2}{3} (\frac{1}{2} \log |\frac{t - 1}{t + 1}|) + C \\ = \frac{1}{3} \log |\frac{\sqrt{1 - x^{3}} - 1}{\sqrt{1 - x^{3}} + 1}| + C . \end{array}$

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