Question

Интеграл sinxdx/(4+cosx)^5

Answer 1

4+cosx=t⇒dt=-sinxdx
$\int\limits {sinx/(4+cosx)^5} \, dx =- \int\limits {1/(4+t)^5} \, dt =1/4(1+y)^4=$$1/4(4+cosx)^4+C$

Answer 2

$\displaystyle \int\frac{\sin(x)\text{d}x}{(4+\cos(x))^5}=\int\frac{-\text{d}(\cos(x))}{(4+\cos(x))^5}=-\int\frac{\text{d}(4+\cos(x))}{(4+\cos(x))^5}=$
$\displaystyle =-\int(4+\cos(x))^{-5}\text{d}(4+\cos(x))=-\frac{(4+\cos(x))^{-4}}{-4}+C=\boxed{\frac{1}{4(4+\cos(x))^4}+C}\phantom{.}.$