Mademoiselle23
contestada

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

[tex] \int { \frac{csc^2(x)}{1+cot(x)} } \, dx [/tex]

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

[tex]= \int{ (\frac{1}{sin^2x} * \frac{1}{1+ \frac{cos(x)}{sinx}} )} \, dx[/tex]

[tex]= \int { \frac{1}{Sin^2(x) + Sin(x)Cos(x)} } \, dx = \int { \frac{1}{Sin(x)}* \frac{1}{Sin(x) + Cos(x)} } \, dx[/tex]

Respuesta :

[tex]\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[/tex]

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