First slide

Methods of integration

Question

If $\int \frac{d x}{x \sqrt{5 x^{2} - 3}} = K \tan^{- 1} f (x) + C$ then

Easy

A

$f (x) = \sqrt{\frac{5}{3} x^{2} - 1}, K = \frac{1}{\sqrt{5}}$
B

$f (x) = \sqrt{\frac{5}{3} x^{2} - 1}, K = \frac{1}{\sqrt{3}}$
C

$f (x) = \frac{1}{2} \sqrt{5 x^{2} - 3}, K = \frac{1}{\sqrt{5}}$
D

none of these

Solution

Putting $5 x^{2} - 3 = t^{2} so 5 x d x = t d t$
$\begin{array}{l} \int \frac{d x}{x \sqrt{5 x^{2} - 3}} = \int \frac{x d x}{x^{2} \sqrt{5 x^{2} - 3}} = \frac{1}{5} \int \frac{5 t d t}{(t^{2} + 3) t} \\ = \int \frac{d t}{t^{2} + 3} = \frac{1}{\sqrt{3}} \tan^{- 1} \frac{t}{\sqrt{3}} + C \\ = \frac{1}{\sqrt{3}} \tan^{- 1} \sqrt{\frac{5 x^{2} - 3}{3}} + C . \end{array}$

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