Question

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


`\int { (csc^2(x))/(1+cot(x)) } \, dx`

This is how I've been trying to solving it, but I'm not sure what to do after some point


`= \int{ ((1)/(sin^2x) * (1)/(1+ (cos(x))/(sinx)) )} \, dx`


`= \int { (1)/(Sin^2(x) + Sin(x)Cos(x)) } \, dx = \int { (1)/(Sin(x))* (1)/(Sin(x) + Cos(x)) } \, dx`

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`