First slide

Methods of integration

Question

$I = \int \cos \{2 \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}}\} d x$ is equal to

Easy

A

$\frac{1}{2} x^{2} + C$
B

$\frac{1}{8} (x^{2} - 1) + C$
C

$\frac{1}{2} x + C$
D

$\frac{1}{4} (x^{2} + 1) + C$

Solution

Put $x = \cos θ \Rightarrow d x = - \sin θ d θ$
$\begin{array}{l} I = \int \cos (2 \tan^{- 1} \tan θ / 2) (- \sin θ) d θ \\ = - \int \cos θ \sin θ d θ \\ = \frac{\cos^{2} θ}{2} + C = \frac{x^{2}}{2} + C \end{array}$

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