int(sqrt(1+cosx)dx =?

প্রশ্ন: \(\int \sqrt{1+\cos x} \, dx = ?\)

সমাধান:

আমরা জানি, \(1 + \cos x = 2\cos^2 \frac{x}{2}\)

অতএব, \(\int \sqrt{1+\cos x} \, dx = \int \sqrt{2\cos^2 \frac{x}{2}} \, dx\)

\(= \int \sqrt{2} \left| \cos \frac{x}{2} \right| \, dx\)

যদি \(\cos \frac{x}{2} \ge 0\) হয়, তবে

\(= \sqrt{2} \int \cos \frac{x}{2} \, dx\)

\(= \sqrt{2} \cdot \frac{\sin \frac{x}{2}}{\frac{1}{2}} + c\)

\(= 2\sqrt{2} \sin \frac{x}{2} + c\)

সুতরাং, \(\int \sqrt{1+\cos x} \, dx = 2\sqrt{2} \sin \frac{x}{2} + c\)

✔️