Question

My attempts at solving this integral keep failing... I'd really appreciate some help :)

Answer 1

$\bf \displaystyle \int \cfrac{csc^2(x)}{1+cot(x)}\cdot dx\\\\ -----------------------------\\\\ u=1+cot(x)\implies \cfrac{du}{dx}=-csc^2(x)\implies \cfrac{du}{-csc^2(x)}=dx\\\\ -----------------------------\\\\ \displaystyle \int\cfrac{csc^2(x)}{u}\cdot \cfrac{du}{-csc^2(x)}\implies -\int \cfrac{1}{u}\cdot du \\\\\\ -ln|u|+C\implies -ln|1+cot(x)|+C$