First slide

Methods of integration

Question

$\int \frac{d x}{x^{\frac{1}{5}} {(4 + x^{\frac{4}{5}})}^{\frac{1}{2}}}$

NA

A

$\sqrt{1 + x^{4 / 5}} + c$
B

$\frac{5}{2} \sqrt{1 + x^{4 / 5}} + c$
C

$x^{4 / 5} {(1 + x^{4 / 5})}^{1 / 2} + c$
D

$\frac{2}{5} \sqrt{1 + x^{4 / 5}} + c$

Solution

$\int \frac{d x}{x^{\frac{1}{5}} {(4 + x^{\frac{4}{5}})}^{\frac{1}{2}}}$ $= \int \frac{5 d t}{4 t^{\frac{1}{2}}} (L e t 4 + x^{\frac{4}{5}} = t \Rightarrow \frac{4 / 5}{x^{1 / 5}} d x = d t)$
$= \frac{5}{4} \sqrt{t} \times 2 + C = \frac{5}{2} \times {(1 + x^{\frac{4}{5}})}^{\frac{1}{2}} + C$

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