Compute the integral
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$$\begin{aligned}{\int }_{0}^{\pi /2}\frac{dx}{a{\cos }^{2}x + b{\sin }^{2}x},\;a,b \gt 0.\tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned}$$
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$$\begin{aligned} \mbox{原式}=&\int_0^\frac{\pi}{2}\frac{\sec^2x\mathrm{ d} x}{a+b\tan^2x} \stackrel{\tan x=t}{=}\int_0^\infty \frac{\mathrm{ d} t}{a+bt^2} \stackrel{\sqrt{b}t=as}{=}\int_0^\infty \frac{\frac{a}{\sqrt{b}}\mathrm{ d} s}{a(1+s^2)} =\frac{1}{\sqrt{ab}}\cdot\frac{\pi}{2}=\frac{\pi}{2\sqrt{ab}}. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{www.zhangzujin.cn}\end{array}}\end{aligned}$$
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