Question

Calcule a integral indefinida de cotg^4(x)*cossec^6(x) dx. Como Resolver?

Answer 1

Levar em consideração a seguinte relação trigonométrica:
$1+\cot^2x=\csc^2x$
$\displaystyle \int\cot^4x\csc^6xdx=\int\cot^4x\csc^4x\csc^2xdx\\\\\int\cot^4x(1+\cot^2x)^2\csc^4xdx\longrightarrow u=\cot x\implies du=-\csc^2xdx\\\\\int u^4(1+u^2)^2-csc^2xdx=\int -u^4(1+u^2)^2du$
então:

$\displaystyle \int -u^4(1+u^2)^2du=\int -u^4(1+2u^2+u^4)du=-\int u^4+2u^6+u^8du\\\\i)~~~~-\int u^4+2u^6+u^8du=-\frac{u^5}{5}-2\frac{u^7}{7}-\frac{u^9}{9}\\\\ii)~~~u=\cot x\rightarrow \boxed{-\frac{1}{5}\cot^5x-\frac{2}{7}\cot^7x-\frac{1}{9}\cot^9x}\\\\\\iii)~~\int \cot^4x\csc^6xdx=\boxed{-\frac{1}{5}\cot^5x-\frac{2}{7}\cot^7x-\frac{1}{9}\cot^9x+C}$