# Integration By Parts

1. Feb 3, 2014

### jdawg

1. The problem statement, all variables and given/known data
∫cosx(lnsinx)dx

2. Relevant equations

3. The attempt at a solution
u=lnsinx dv=cosxdx
du=cosx/sinx dx v=sinx

=(lnsinx)(sinx)-∫(sinx)(cosx/sinx)dx
=(lnsinx)(sinx)-(sinx)+C

I thought that I did this correctly, but my teacher said that u should equal sinx. Why would u not equal lnsinx?

2. Feb 3, 2014

### jackarms

If you differentiate, you get the original function, so you did it correctly. I think your teacher was saying u should equal sinx if you're doing a simple u-sub to integrate, since that method works as well. You couldn't have u be sinx for an integration by parts, since it isn't a complete term in the integrand -- it's only part of the natural log.

3. Feb 3, 2014

### Dick

You did do it correctly. I think your teacher might be suggesting you do a u-substitution first and then integrate log(u) by parts. I think that's actually a little more complicated, not easier.

4. Feb 3, 2014

### jdawg

Ohh ok! I didn't know you could use u substitution on that one. Thanks for clearing that up :)

5. Feb 3, 2014

### Dick

You can do a u substitution, but then you are left with log(u), which you then need to integrate by parts. Unless you've memorized the integral of log(u). I think your way is better.