Question

Найдите неопределённый интеграл подробно:

Answer 1

$\int\sin5x\sin3xdx=\int\left(\frac{\cos(5x-3x)-\cos(5x+3x)}2\right)dx=\frac12\int\left(\cos2x-\cos8x\right)dx=\\\\=\frac12\int\cos2xdx-\frac12\int\cos8xdx=\frac12\cdot\frac12\sin2x-\frac12\cdot\frac18\sin8x+C=\frac12\sin2x-\frac1{16}\sin8x+C$

$\int\sin^2xdx=\int\frac{1-\cos2x}2dx=\frac12\int dx-\frac12\int\cos2xdx=\frac12x-\frac12\cdot\frac12\sin2x+C=\\=\frac12x-\frac14\sin2x+C$