int_0^(pi/2)sin^5xcosxdx=? 

Explanation:

Another Explanation (5): সমাধান: ধরি, \(I = \int_0^{\frac{\pi}{2}} \sin^5x \cos x \, dx\) এখানে, \(\sin x = z\) ধরলে, \(\cos x \, dx = dz\) হবে। সুতরাং, যখন \(x = 0\), তখন \(z = \sin 0 = 0\) এবং যখন \(x = \frac{\pi}{2}\), তখন \(z = \sin \frac{\pi}{2} = 1\) তাহলে, সমাকলনটি হবে: \(I = \int_0^1 z^5 \, dz\) \(I = \left[ \frac{z^6}{6} \right]_0^1\) \(I = \frac{1^6}{6} - \frac{0^6}{6}\) \(I = \frac{1}{6} - 0\) \(I = \frac{1}{6}\) 🎉 সুতরাং, \(\int_0^{\frac{\pi}{2}} \sin^5x \cos x \, dx = \frac{1}{6}\) 😎