int_0^(1/2) (1)/sqrt(1-3x^2) dx =i  হয়, তবে i=?

প্রশ্ন: \( \int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1-3x^2}} dx = i \) হলে, \( i = ? \)

সমাধান:

ধরি, \( \sqrt{3}x = \sin{\theta} \)
সুতরাং, \( x = \frac{\sin{\theta}}{\sqrt{3}} \)
\( dx = \frac{\cos{\theta}}{\sqrt{3}} d\theta \)

যখন \( x = 0 \), তখন \( \sin{\theta} = 0 \Rightarrow \theta = 0 \)
যখন \( x = \frac{1}{2} \), তখন \( \sin{\theta} = \frac{\sqrt{3}}{2} \Rightarrow \theta = \frac{\pi}{3} \)

অতএব, \( i = \int_{0}^{\frac{\pi}{3}} \frac{1}{\sqrt{1 - \sin^2{\theta}}} \cdot \frac{\cos{\theta}}{\sqrt{3}} d\theta \)
\( = \int_{0}^{\frac{\pi}{3}} \frac{1}{\cos{\theta}} \cdot \frac{\cos{\theta}}{\sqrt{3}} d\theta \)
\( = \frac{1}{\sqrt{3}} \int_{0}^{\frac{\pi}{3}} d\theta \)
\( = \frac{1}{\sqrt{3}} [\theta]_{0}^{\frac{\pi}{3}} \)
\( = \frac{1}{\sqrt{3}} \left( \frac{\pi}{3} - 0 \right) \)
\( = \frac{\pi}{3\sqrt{3}} \)

সুতরাং, \( i = \frac{\pi}{3\sqrt{3}} \). 🎉