Solving the Integral of cos x ln x: Step-by-Step Guide and Tips

[sagita]

Please, how can I solve this?

∫ cos x ln x dx

I get this:

ln x sin x - ∫sin x/x dx

but how do I continue from here?

Thanks in advance...

 

[Mystic998]

The antiderivative of sin(x)/x isn't expressible in terms of elementary functions so perhaps it would be better to change the role of u and dv in your integration by parts.

Edit: At least, I think that's the case. Couldn't hurt to try anyway.

 

[Hurkyl]

That won't change anything -- one cannot be expressed in terms of elementary functions iff the other cannot be expressed as well.

 

[fizzzzzzzzzzzy]

look at it as an equation, and you need to integrate by parts at least twice

 

[dfx]

look at it as an equation, and you need to integrate by parts at least twice

No, this won't help. Even Wolfram gives an answer with Si(x) in it - the integral of Sinx/x.

 

[Vagrant]

So, can't sinx/x be integrated?

 

[HallsofIvy]

Yes, of course it can- its integral is Si(x)! It cannot, however, be integrated in terms of elementary functions.

 

[sagita]

It seems difficult to continue from ln x sin x - ∫sin x/x dx ...

Thanks to averyone who posted. I'll tell you if something different appears.

Thanks again.

 

[Data]

It is impossible to continue without introducing "special" functions or series expansions (from which you won't be able to obtain closed forms). So, play with it for a while, but don't spend too much time on it .

 

[Gib Z]

If you really don't want Si(x), here's your only alternative:

\frac{\sin x}{x} = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n+1)!}.

Integrate that, and there you go.

 

[dimensionless]

And don't forget this one:
\frac{sin(x)}{x} = sinc(x)