int_0^(pi/2)cos^2thetacosthetad theta=?

Explanation:

Another Explanation (5): সমাধান: আমরা \(\int_0^{\frac{\pi}{2}} \cos^2(\theta) \cos(\theta) \, d\theta\) এর মান নির্ণয় করব। ধরি, \(I = \int_0^{\frac{\pi}{2}} \cos^3(\theta) \, d\theta\) \(I = \int_0^{\frac{\pi}{2}} \cos^2(\theta) \cos(\theta) \, d\theta\) \(I = \int_0^{\frac{\pi}{2}} (1 - \sin^2(\theta)) \cos(\theta) \, d\theta\) ধরি, \(u = \sin(\theta)\), সুতরাং \(du = \cos(\theta) \, d\theta\) যখন \(\theta = 0\), তখন \(u = \sin(0) = 0\) যখন \(\theta = \frac{\pi}{2}\), তখন \(u = \sin(\frac{\pi}{2}) = 1\) সুতরাং, \(I = \int_0^1 (1 - u^2) \, du\) \(I = \int_0^1 1 \, du - \int_0^1 u^2 \, du\) \(I = [u]_0^1 - [\frac{u^3}{3}]_0^1\) \(I = (1 - 0) - (\frac{1^3}{3} - \frac{0^3}{3})\) \(I = 1 - \frac{1}{3}\) \(I = \frac{3 - 1}{3}\) \(I = \frac{2}{3}\) অতএব, \(\int_0^{\frac{\pi}{2}} \cos^2(\theta) \cos(\theta) \, d\theta = \frac{2}{3}\) 🎉🎉