Интеграл sinxdx/(4+cosx)^5

4+cosx=t⇒dt=-sinxdx[latex] intlimits {sinx/(4+cosx)^5} , dx =- intlimits {1/(4+t)^5} , dt =1/4(1+y)^4=[/latex][latex]1/4(4+cosx)^4+C[/latex]

[latex]displaystyle intfrac{sin(x)ext{d}x}{(4+cos(x))^5}=intfrac{-ext{d}(cos(x))}{(4+cos(x))^5}=-intfrac{ext{d}(4+cos(x))}{(4+cos(x))^5}=[/latex][latex]displaystyle =-int(4+cos(x))^{-5}ext{d}(4+cos(x))=-frac{(4+cos(x))^{-4}}{-4}+C=oxed{frac{1}{4(4+cos(x))^4}+C}phantom{.}.[/latex]