# Integral of cos and ln

1. Apr 8, 2007

### 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?

2. Apr 8, 2007

### 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.

3. Apr 8, 2007

### Hurkyl

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

4. Apr 8, 2007

### fizzzzzzzzzzzy

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

5. Apr 9, 2007

### dfx

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

6. Apr 9, 2007

### Vagrant

So, can't sinx/x be integrated?

7. Apr 9, 2007

### HallsofIvy

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

8. Apr 9, 2007

### 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.

9. Apr 9, 2007

### 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 .

10. Apr 10, 2007

### Gib Z

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

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

Integrate that, and there you go.

11. Apr 10, 2007

### dimensionless

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