int(cos^3xdx=?

আমরা \( \int \cos^3 x \, dx \) এর মান নির্ণয় করব।

আমরা জানি, \( \cos 3x = 4\cos^3 x - 3\cos x \)।

সুতরাং, \( \cos^3 x = \frac{1}{4} (\cos 3x + 3\cos x) \)। 😃

তাহলে,

\[ \begin{aligned} \int \cos^3 x \, dx &= \int \frac{1}{4} (\cos 3x + 3\cos x) \, dx \\ &= \frac{1}{4} \int (\cos 3x + 3\cos x) \, dx \\ &= \frac{1}{4} \left( \int \cos 3x \, dx + 3 \int \cos x \, dx \right) \\ &= \frac{1}{4} \left( \frac{\sin 3x}{3} + 3\sin x \right) + C \\ &= \frac{1}{12} \sin 3x + \frac{3}{4} \sin x + C \\ &= \frac{1}{12} \sin 3x + \frac{9}{12} \sin x + C \\ &= \frac{1}{12} (\sin 3x + 9\sin x) + C \end{aligned} \]

অতএব, \( \int \cos^3 x \, dx = \frac{1}{12} (\sin 3x + 9\sin x) + C \)। 🎉