গণিত প্রশ্ন

AI-তৈরি নমুনা উত্তর

প্রদত্ত সমাকলন:
$I = \int_{2}^{3} \frac{dx}{(x - 1) \sqrt{x^2 - 2x}}$
$= \int_{2}^{3} \frac{dx}{(x - 1) \sqrt{(x - 1)^2 - 1}}$

ধরি, $x - 1 = \sec{\theta}$
$\implies dx = \sec{\theta}\tan{\theta} d\theta$

সীমা পরিবর্তন:

• যখন $x = 2$, $\sec{\theta} = 2 - 1 = 1 \implies \theta = 0$
• যখন $x = 3$, $\sec{\theta} = 3 - 1 = 2 \implies \theta = \frac{\pi}{3}$

অতএব,
$I = \int_{0}^{\frac{\pi}{3}} \frac{\sec{\theta}\tan{\theta}\, d\theta}{\sec{\theta}\sqrt{\sec^2{\theta} - 1}}$
$= \int_{0}^{\frac{\pi}{3}} \frac{\sec{\theta}\tan{\theta}\, d\theta}{\sec{\theta}\tan{\theta}}$
$= \int_{0}^{\frac{\pi}{3}} d\theta$
$= [\theta]_{0}^{\frac{\pi}{3}} = \frac{\pi}{3} - 0 = \frac{\pi}{3}$

উত্তর: $\frac{\pi}{3}$