Evalute : 1 a^2-b^2 x^2 d x - Vidyadip

Question

Evalute : $\int \frac{1}{a^2-b^2 x^2} d x$

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Answer

$
\begin{aligned}
\int \frac{1}{a^2-b^2 x^2} d x=\frac{1}{b^2} \int \frac{1}{\frac{a^2}{b^2}-x^2} d x\\
=\frac{1}{b^2} \int \frac{1}{\left(\frac{a}{b}\right)^2-x^2} d x \\
=\frac{1}{b^2} \times \frac{1}{2\left(\frac{a}{b}\right)} \log \left|\frac{\frac{a}{b}+x}{\frac{a}{b}-x}\right|+c \\
=\frac{1}{2 a b} \log \left|\frac{a+b x}{a-b x}\right|+c .
\end{aligned}
$

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