Antiderivative (sin x)(ln x)

**allyall** · October 4th 2008, 07:30 PM

I was asked for help on finding the anti derivative of sin x multiplied by ln x. I have tried a few things doing it by parts and end up with the antiderivative of sin x / x in the answer, which isn't possible if it isn't on an interval, the other way seemed to get more complicated instead of simpler so I didn't think that worked.

I can't think of a simple substitution that would work, trig has never been a strong suit of mine, so I'm wondering what I'm missing, wanting to know how to do the problem if its possible.

Thanks in advance.

**Fire Mage** · October 4th 2008, 08:02 PM

Well when I started doing it by parts the first thing I got was:

$-Lnxcosx + \int(cosx)(1/x)dx$

Which I really had no idea where to go with. When I used the Wolfram integrator it had:

$Ci(x) - lnxcosx$

Where it said Ci(x) was "CosineIntegral" which was typically found as:

$-\int_{x}^{\infty}\frac{cost}{t}dt$

So I don't really know. I haven't learned such a thing as that yet.

**mr fantastic** · October 4th 2008, 09:55 PM

Originally Posted by allyall

I was asked for help on finding the anti derivative of sin x multiplied by ln x. I have tried a few things doing it by parts and end up with the antiderivative of sin x / x in the answer, which isn't possible if it isn't on an interval, the other way seemed to get more complicated instead of simpler so I didn't think that worked.

I can't think of a simple substitution that would work, trig has never been a strong suit of mine, so I'm wondering what I'm missing, wanting to know how to do the problem if its possible.

Thanks in advance.

As implied by the previously poster, there is no answer in terms of a finite number of elementary functions. Why is the person who asked you for help trying to find it?

**allyall** · October 5th 2008, 08:07 PM

It was on their homework. I have a degree in math ed (though I'm not teaching currently) and often help people I know with stuff.

I suspected it wasn't possible but wanted to be sure. I found this site thru a google search, it turned up for a question on the integral of sin(ln x).

Thanks for the replies.

**mr fantastic** · October 5th 2008, 11:47 PM

Originally Posted by allyall

It was on their homework. [snip]

!!!

Was it a definite integral? Perhaps to be done numerically ....? What subject is the homework from? What level at university - 2nd year, perhaps? I'm astounded, as you can possibly guess lol!

Perhaps there is a typo in the homework ....?

**shawsend** · October 6th 2008, 04:07 AM

Originally Posted by allyall

I was asked for help on finding the anti derivative of sin x multiplied by ln x.

Put an end to special function inequality: equal rights for special functions.

$\int \sin(x)\ln(x)dx=\textbf{CosI}(x)-\cos(x)\ln(x)$

No?

How about:

$\frac{d}{dx}\Biggr(\textbf{CosI}(x)-\cos(x)\ln(x)\Biggr)=\frac{d}{dx}\left(-\int_x^{\infty}\frac{\cos(t)}{t}dt-\cos(x)\ln(x)\right)$

$=-\left(0-\frac{\cos(x)}{x}\right)-\left(\frac{\cos(x)}{x}-\sin(x)\ln(x)\right)$

$=\sin(x)\ln(x)$

Works for me.

**allyall** · October 6th 2008, 08:13 PM

Perhaps there is a typo in the homework ....?

That is my thought, though perhaps it was not. 1st year college, 2nd semester calculus.

Thank you shawsend for showing it is possible/how cosine integral works. Not something I've played with much, appreciate seeing it though!

**mr fantastic** · October 6th 2008, 08:31 PM

Originally Posted by allyall

That is my thought, though perhaps it was not. 1st year college, 2nd semester calculus.

Thank you shawsend for showing it is possible/how cosine integral works. Not something I've played with much, appreciate seeing it though!

I doubt a student in 1st year college 2nd semester calculus would be expected to know about the cosine integral.

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