Question

\[\int\limits_0^\infty \frac{1}{a^2 + b^2 x^2} dx\]

Answer 1

\[Let\ I = \int_0^\infty \frac{1}{a^2 + b^2 x^2} d x\ . Then, \]
\[I = \frac{1}{a^2} \int_0^\infty \frac{1}{1 + \frac{b^2 x^2}{a^2}} d x\]
\[ \Rightarrow I = \frac{1}{a^2} \int_0^\infty \frac{1}{1 + \left( \frac{bx}{a} \right)^2} d x\]
\[ \Rightarrow I = \frac{a}{b a^2} \left[ \tan^{- 1} \left( \frac{bx}{a} \right) \right]_0^\infty \]
\[ \Rightarrow I = \frac{1}{ab}\left( \tan^{- 1} \infty - \tan^{- 1} 0 \right)\]
\[ \Rightarrow I = \frac{\pi}{2ab}\]