First slide

Methods of integration

Question

The value of $\int \frac{d x}{\sqrt{4 x - 3 - x^{2}}}$ is equal to

Easy

A

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

$\log |(x - 2) + \sqrt{4 x - 3 - x^{2}}|$
C

$\log |(x - 1) + \sqrt{4 x - 3 - x^{2}}| + C$
D

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

Solution

$\sqrt{4 x - 3 - x^{2}} = \sqrt{- (x^{2} - 4 x + 3)}$
$= \sqrt{- ((x - 2)^{2} - 1)} = \sqrt{1 - (x - 2)^{2}}$
So, $\int \frac{d x}{\sqrt{4 x - 3 - x^{2}}} = \int \frac{d x}{\sqrt{1 - (x - 2)^{2}}} = \sin^{- 1} (x - 2) + C$

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