How to integrate int sin(ln(x)) ?

[Shaybay92]

Homework Statement

I am having trouble solving the following:

$$\int$$sin(lnx) dx

The Attempt at a Solution

I let u = ln x but this makes xdu = dx so I am left with

$$\int$$x sinu du

 

[Cyosis]

That's the substitution you want to use,but you can't leave the x in there. If u=lnx, then x is?

 

[Shaybay92]

Would x = u²?

$$\int$$u^e sin u du

Then do you do it by parts or something?

 

[Cyosis]

How on Earth did you get that? Can you solve $u=\ln x$ for x?

 

[Shaybay92]

Oops I meant e^u, is that right?

 

[Cyosis]

Yes that's correct. You can now use integration by parts, twice.

 

[Shaybay92]

does e(lnx) cancel to 1?

 

[Cyosis]

No and you should really know that at this stage! If a function has an inverse, $f^{-1}(x)$, then $f^{-1}(f(x))=f(f^{-1}(x))=x$. So what is $e^{\ln x}$ and $\ln e^x$?