Question

Как решить интеграл cos^4x/sinx?

Answer 1

$int frac{cos^4 x}{sin x} dx=\\-int frac{cos^4 x (-sin x) dx }{sin^ 2x}=\\-int frac{cos^4 x d (cos x)}{1-cos^ 2x}=|cos x=t|=\\int frac{-t^4 dt}{1-t^2}=\\int frac{(1-t^4-1)dt}{1-t^2}=\\int frac{(1-t^2)(1+t^2) dt}{1-t^2}-frac{dt}{1-t^2}=\\int(1+t^2)dt-intfrac{dt}{1-t^2}=t+frac{t^3}{3}-frac{1}{2}ln|frac{1+t}{1-t}}|+C=\\cos x+frac{cos^3 x}{3}-frac{1}{2}ln|frac{1+cos x}{1-cos x}|+C$