Question

Помогите найти интеграл

Answer 1

$\int \frac{\sin^3x dx}{\sqrt{\cos x}} = -\int \frac{\sin^2x d( \cos x )}{\sqrt{ \cos x}} = \int \frac{(\cos^2x-1) d( \cos x )}{\sqrt{ \cos x}} = \left[\begin{array}{ccc}\cos x = t\end{array}\right] =\\ = \int \frac{t^2-1}{\sqrt{t}} dt = \int t^{\frac{3}{2}} dt - \int t^{-\frac{1}{2}} dt = \frac{2}{5} t^{\frac{5}{2}} -2t^{\frac{1}{2}} +C = \frac{2}{5} t^2\sqrt{t} - 2\sqrt{t} +C = \\ = \frac{2}{5} \cos^2x \sqrt{ \cos x} - 2\sqrt{ \cos x} +C$