int_0^(pi/4)cos^2xdx এর মান হলো-

Explanation:

Another Explanation (5): সমাধান: আমরা জানি, \( \cos 2x = 2\cos^2 x - 1 \)। সুতরাং, \( \cos^2 x = \frac{1 + \cos 2x}{2} \) অতএব, \( \int_0^{\pi/4} \cos^2 x \, dx = \int_0^{\pi/4} \frac{1 + \cos 2x}{2} \, dx \) \( = \frac{1}{2} \int_0^{\pi/4} (1 + \cos 2x) \, dx \) \( = \frac{1}{2} \left[ x + \frac{\sin 2x}{2} \right]_0^{\pi/4} \) \( = \frac{1}{2} \left[ \left( \frac{\pi}{4} + \frac{\sin (\pi/2)}{2} \right) - \left( 0 + \frac{\sin 0}{2} \right) \right] \) \( = \frac{1}{2} \left[ \frac{\pi}{4} + \frac{1}{2} - 0 \right] \) \( = \frac{\pi}{8} + \frac{1}{4} \) সুতরাং, \( \int_0^{\pi/4} \cos^2 x \, dx = \frac{\pi}{8} + \frac{1}{4} \) 🥳