\[
\begin{array}{l}
\text { Given that, } \int_{0}^{\pi / 2} \frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x} d x \\
=\int_{0}^{\pi / 2} \frac{1}{\frac{\cos ^{2} x}{\sin ^{2} x}+b^{2} \frac{\cos ^{2} x}{\cos ^{2} x} d x} \\
=\int_{0}^{\pi / 2} \frac{\sec ^{2} x}{a^{2} \tan ^{2} x+b^{2}} d x
\end{array}
\]
Let \( \tan \mathrm{x}=t \) then, \( d t=\sec ^{2} x d x \) and limits change from \( 0 \rightarrow 0, \frac{\Pi}{2} \rightarrow \infty \)
\[
\begin{array}{l}
\frac{1}{a^{2}} \int_{0}^{\infty} \frac{d t}{1+\left(\frac{b}{a}\right)^{2} t^{2}} \\
=\frac{1}{a^{2}} \cdot \frac{1}{(b / a)} \cdot\left(\tan ^{-1}\left(\frac{t}{b / a}\right)\right) \\
=\frac{1}{a b}\left(\frac{\Pi}{2}-0\right)=\frac{\Pi}{2 a b}
\end{array}
\]