\( \int_{0}^{\frac{\pi}{2}} sin^5\theta cos\theta d\theta \) = ?

প্রশ্ন: \( \int_{0}^{\frac{\pi}{2}} sin^5\theta cos\theta d\theta \) = ?

সমাধান:

ধরি, \( u = sin\theta \)
সুতরাং, \( du = cos\theta d\theta \)

যখন \( \theta = 0 \), তখন \( u = sin(0) = 0 \)
যখন \( \theta = \frac{\pi}{2} \), তখন \( u = sin(\frac{\pi}{2}) = 1 \)

অতএব, ইন্টিগ্রালটি হবে:

\( \int_{0}^{1} u^5 du \)

আমরা জানি, \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)

সুতরাং, \( \int_{0}^{1} u^5 du = \left[ \frac{u^{5+1}}{5+1} \right]_{0}^{1} = \left[ \frac{u^6}{6} \right]_{0}^{1} \)

\( = \frac{1^6}{6} - \frac{0^6}{6} = \frac{1}{6} - 0 = \frac{1}{6} \)

সুতরাং, \( \int_{0}^{\frac{\pi}{2}} sin^5\theta cos\theta d\theta = \frac{1}{6} \) 🥳