First slide

Methods of integration

Question

The value of $\int \frac{d x}{\sqrt[4]{(x - 1)^{3} (x + 2)^{5}}}$ is

Moderate

A

$3 (x - 1)^{1 / 4} - (5 / 3) (x + 2)^{3 / 4} + C$
B

$4 (\sqrt[4]{x - 1}) - 5 (\sqrt[4]{(x + 2)^{5}}) + C$
C

$\frac{4}{3} \sqrt[4]{\frac{x - 1}{x + 2} + C}$
D

none of these

Solution

$\begin{array}{l} I = \int \frac{d x}{4 \sqrt{(x - 1)^{3} (x + 2)^{5}}} \\ = \int {(\frac{x - 1}{x + 2})}^{- 3 / 4} \frac{1}{(x + 2)^{2}} d x \end{array}$
Now put $\frac{x - 1}{x + 2} = t$ to obtain
$I = \frac{1}{3} \int t^{- 3 / 4} d t = \frac{4}{3} t^{1 / 4} + C$

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