int sin^2 (x/2) cos (x/2)dx = ?

A. 2sinx+c

(sin(x/2))^2 +c

2sin^3(x/2) +c

2/3 sin^3(x/2) +c

সঠিক উত্তরঃ D.

2/3 sin^3(x/2) +c

Explanation:

Another Explanation (5): ```html

ধরি, \(u = \sin(\frac{x}{2})\) 😊

তাহলে, \(\frac{du}{dx} = \cos(\frac{x}{2}) \cdot \frac{1}{2}\)

সুতরাং, \(dx = \frac{2 du}{\cos(\frac{x}{2})}\) 🎉

এখন, \(\int \sin^2(\frac{x}{2}) \cos(\frac{x}{2}) dx = \int u^2 \cos(\frac{x}{2}) \cdot \frac{2 du}{\cos(\frac{x}{2})}\)

=\( 2 \int u^2 du \) 🤩

=\( 2 \cdot \frac{u^3}{3} + c \) ✨

=\( \frac{2}{3} u^3 + c \) 🎈

=\( \frac{2}{3} \sin^3(\frac{x}{2}) + c \) ✅

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