Provar a integral de cossecante ao quadrado

[dressa_mwar1]

\(\int cossec^2 x dx\)

[danko71]

\(Seja\ y=cotg(x)=\frac{cos(x)}{sen(x)}\\y'=\frac{v.u'-uv'}{v^2}(1)\\Sendo\ u=cos(x)\rightarrow u'=-sen(x);v=sen(x)\rightarrow v'=cos(x)\ e\ v^2=sen^2(x)\\Substituindo\ em\ (1)\\ y'=\frac{sen(x)(-sen(x))dx-cos(x).cos(x)dx}{sen^2(x)}\\ =\frac{-sen^2(x)-cos^2(x)}{sen^2(x)}\\ =-\frac{(sen^2(x)+cos^2(x))}{sen^2(x)}=-\frac{1}{sen^2(x)}=-cossec^2(x)dx\\ \int cossec^2(x)dx=-cotg(x)\)