Question

Вычислить определенный интеграл

Answer 1

Ответ:

$\int\limits^{ \frac{\pi}{2} } _ {0} \sin {}^{2} (x) \cos {}^{3} (x)dx = \int\limits^{ \frac{\pi}{2} } _ {0} \sin {}^{2} (x) \times \cos {}^{2} (x) \cos(x) dx = \\ = \int\limits^{ \frac{\pi}{2} } _ {0}(1 - \sin {}^{2} (x)) \times \sin {}^{2} (x)) d( \sin(x)) = \\ = \int\limits^{ \frac{\pi}{2} } _ {0} \sin {}^{2} (x) d( \sin(x)) - \int\limits^{ \frac{\pi}{2} } _ {0} \sin {}^{4} (x) d(\sin(x) ) = \\ = ( \frac{ \sin {}^{3} (x) }{3} - \frac{ \sin {}^{5} (x) }{5} ) | ^{ \frac{\pi}{2} } _ {0} = \frac{1}{3} - \frac{1}{5} - 0 - 0 = \frac{2}{15}$