int_0^(π/2)cos^2dx =?

প্রশ্ন: \( \int_{0}^{\frac{\pi}{2}} \cos^2(x) \, dx = ? \)

সমাধান:

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

সুতরাং, \( \int_{0}^{\frac{\pi}{2}} \cos^2(x) \, dx = \int_{0}^{\frac{\pi}{2}} \frac{1 + \cos(2x)}{2} \, dx \)

= \( \frac{1}{2} \int_{0}^{\frac{\pi}{2}} (1 + \cos(2x)) \, dx \)

= \( \frac{1}{2} \left[ \int_{0}^{\frac{\pi}{2}} 1 \, dx + \int_{0}^{\frac{\pi}{2}} \cos(2x) \, dx \right] \)

= \( \frac{1}{2} \left[ x \Big|_0^{\frac{\pi}{2}} + \frac{\sin(2x)}{2} \Big|_0^{\frac{\pi}{2}} \right] \)

= \( \frac{1}{2} \left[ \left( \frac{\pi}{2} - 0 \right) + \frac{1}{2} \left( \sin(\pi) - \sin(0) \right) \right] \)

= \( \frac{1}{2} \left[ \frac{\pi}{2} + \frac{1}{2} (0 - 0) \right] \)

= \( \frac{1}{2} \cdot \frac{\pi}{2} \)

= \( \frac{\pi}{4} \)

অতএব, \( \int_{0}^{\frac{\pi}{2}} \cos^2(x) \, dx = \frac{\pi}{4} \)

উত্তর: \( \frac{\pi}{4} \) 🎉