Question

Найдите общий вид первообразных для функции f(x)= sin^4 х

Answer 1

 

$int sin^4x dx = int (sin^2x)(1-cos^2x) dx =\\ int sin^2x dx -int sin^2xcos^2x dx =\\ int frac{1-cos2x}{2} dx -int frac{1}{4}sin^22x dx =\\ int frac{1-cos2x}{2} dx -int frac{1}{4}frac{1-cos4x}{2} dx =\\ frac{x}{2} - frac{1}{4}sin2x - frac{x}{8} + frac{1}{32}sin4x + C =\\ frac{3x}{8} - frac{1}{4}sin2x + frac{1}{32}sin4x + C$