Question

\[\int\frac{1}{a^2 x^2 + b^2} dx\]

Answer 1

\[\int\frac{dx}{a^2 x^2 + b^2}\]
\[ = \frac{1}{a^2}\int\frac{dx}{x^2 + \left( \frac{b}{a} \right)^2} \]
\[ = \frac{1}{a^2} \times \frac{a}{b} \tan^{- 1} \left( \frac{x}{\frac{b}{a}} \right) + C \left[ \therefore \int\frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{- 1} \left( \frac{x}{a} \right) + C \right]\]
\[ = \frac{1}{ab} \tan^{- 1} \left( \frac{ax}{b} \right) + C\]