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Math Help - Antiderivative (sin x)(ln x)

  1. #1
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    Antiderivative (sin x)(ln x)

    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.
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  2. #2
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    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.
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  3. #3
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    Quote Originally Posted by allyall View Post
    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?
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  4. #4
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    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.
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  5. #5
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    Quote Originally Posted by allyall View Post
    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 ....?
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  6. #6
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    Quote Originally Posted by allyall View Post
    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.
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  7. #7
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    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!
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  8. #8
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    Quote Originally Posted by allyall View Post
    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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