First slide

Evaluation of definite integrals

Question

$\int \frac{\tan (\sin^{- 1} x)}{\sqrt{1 - x^{2}}} d x$ =

Easy

A

$\frac{\tan^{2} (\sin^{- 1} x)}{2} + C$
B

$\log | (\sin^{- 1} x) | + C$
C

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

$\log | 1 + x^{2} | + C$

Solution

$\sin^{- 1} x = t \Rightarrow \frac{1}{\sqrt{1 - x^{2}}} d x = d t$
$\int \tan t d t = \log | \sec t | + C$

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

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