Evaluate the following integral: 1 a^2x^2-b^2 dx - Vidyadip

Question

Evaluate the following integral:
$\int\frac{1}{\text{a}^2\text{x}^2-\text{b}^2}\text{ dx}$

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Answer

Let $\text{I}=\int\frac{1}{\text{a}^2\text{x}^2-\text{b}^2}\text{ dx}$
$=\frac{1}{\text{a}^2}\int\frac{1}{\text{x}^2-\frac{\text{b}^2}{\text{a}^2}}\text{ dx}$
$=\frac{1}{\text{a}^2}\int\frac{1}{\text{x}^2-\big(\frac{\text{b}}{\text{a}}\big)^2}\text{ dx}$
$\text{I}=\frac{1}{\text{a}^2}\times\frac{1}{2\times\big(\frac{\text{b}}{\text{a}}\big)}\times\Bigg|\log\frac{\text{x}-\frac{\text{b}}{\text{a}}}{\text{x}+\frac{\text{b}}{\text{a}}}\Bigg|+\text{C}$ $\Big[$Since $\int\frac{1}{\text{x}^2-\text{a}^2}\text{ dx}=\frac{1}{2\text{a}}\log\Big|\frac{\text{x}-\text{a}}{\text{x}+\text{a}}\Big|+\text{C}\Big]$
$\text{I}=\frac{1}{2\text{ab}}\log\Big|\frac{\text{ax}-\text{b}}{\text{ax}+\text{b}}\Big|+\text{C}$

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