Integrating cos^2(x)/sinx using substitution and integration by parts

[shn]

i have try to intergral cos^2( x)/sinx. When i used sinx=t i got {[(1-t^2)^n-(1/2)]\t}. When i use intergral by parts i got {cos^2n-1(x)[1-cos^2(x)]} to intergral. If you could give me a tip to intergral this i would bn thankful to you!

 

[Clever-Name]

Where did the n come from? There is no n in the initial question.

If I were you I would start by writing this as:

\int cot(x)cos(x)dx

What identities do you know for cot(x) that might make this easier?

edit - Your post changed during the time I was writing this. Yes, the way it is now written is actually easier to see the next step.

 

[shn]

thankz man!i did'nt read it correctly!:'(

 

[Mark44]

i have try to intergral cos^2( x)/sinx. When i used sinx=t i got {[(1-t^2)^n-(1/2)]\t}. When i use intergral by parts i got {cos^2n-1(x)[1-cos^2(x)]} to intergral. If you could give me a tip to intergral this i would bn thankful to you!

First off, there is no such word in English as "intergral."

And you don't "integral" something - you integrate it.

Regarding your answers, n should not appear in them, so you are not using integration tables correctly, assuming that's what you're doing.

The simplest way to approach this problem is to replace cos²(x) by 1 - sin²(x), which changes the integral you started with to
$$ \int \frac{1 - sin^2(x)}{sin(x)}~dx$$

Split this into two integrals and the rest is pretty straightforward.