First slide

Methods of integration

Question

Evaluate $\int x^{\frac{- 2}{3}} {(1 + x^{2 / 3})}^{- 1}$ .

Moderate

A

$\tan^{- 1} (x^{1 / 3}) + C$
B

$\tan^{- 1} (x^{2 / 3}) + C$
C

$3 \tan^{- 1} (x^{1 / 3}) + C$
D

None of these

Solution

Here, P = -1, is a negative interger and m and n are rational numbers.
Put $x = t^{3} \Rightarrow dx = 3 t^{2} dt$
So, $I = \int t^{- 2} {(1 + t^{2})}^{- 1} 3 t^{2} dt = \int \frac{3 dt}{1 + t^{2}} = 3 \tan^{- 1} (x^{1 / 3}) + C$

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