First slide

Methods of integration

Question

$\int x^{5} \sqrt[3]{{(1 + x^{3})}^{2}} d x$ is equal to

Easy

A

$\frac{1}{8} u^{8 / 3} - \frac{1}{5} u^{5 / 3} + C, u = 1 + x^{3}$
B

$\frac{3}{8} u^{8 / 3} - \frac{3}{5} u^{5 / 3} + C, u = 1 + x^{3}$
C

$- \frac{3}{8} u^{8 / 3} + \frac{3}{5} u^{5 / 3} + C, u = 1 + x^{3}$
D

$- \frac{1}{8} u^{8 / 3} + \frac{1}{5} u^{5 / 3} + C, u - 1 + x^{3}$

Solution

Put $1 + x^{3} = t^{3} \Rightarrow x^{2} d x = t^{2} d t$
$\int x^{5} \sqrt[3]{{(1 + x^{3})}^{2}} d x = \int (t^{3} - 1) t^{4} d t$
$\begin{array}{l} = \frac{t^{8}}{8} - \frac{t^{5}}{5} + C \\ = \frac{1}{8} {(1 + x^{3})}^{8 / 3} - \frac{1}{5} {(1 + x^{3})}^{5 / 3} + C . \end{array}$

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