A=acos² $\theta$ +bsin² $\theta$  

  $\int_{0}^{\frac{\pi}{2}} \frac{d \theta}{A}$ এর মান কোনটি?

$\begin{array}{l} \text { } \int_{0}^{\frac{\pi}{2}} \frac{\mathrm{d} \theta}{\mathrm{acos}^{2} \theta+\mathrm{b} \sin ^{2} \theta} \\ =\int_{0}^{\frac{\pi}{2}} \frac{\sec ^{2} \theta \mathrm{d} \theta}{\mathrm{a}+\mathrm{btan}^{2} \theta} \end{array}$

ধরি, $\tan \theta=\mathrm{z}$

$\begin{array}{l} \therefore \sec ^{2} \theta d \theta=d z \\ \text { এখন, } \int_{0}^{\infty} \frac{d z}{a+b z^{2}} \\ =\left.\left[\frac{1}{\sqrt{a b}} \tan ^{-1}\left(\sqrt{\frac{b}{a}} z\right)\right]_{0}^{\infty}\right|_{\text {তখन } \theta=0 \text { তখন } z=0} \theta=\frac{\pi}{2} \text { তখन } z=\infty \\ =\frac{1}{\sqrt{a b}} \times \frac{\pi}{2}=\frac{\pi}{2 \sqrt{a b}} \end{array}$