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Homework Help: An indefinite integral

  1. Jan 7, 2010 #1
    Hi,

    Looking to integrate the indefinite integral:

    [tex]\int tan~x\cdot ln~x\cdot cos~x[/tex]

    Since tan x = sin x/ cos x, this integral be written as [tex]\int sin~x\cdot ln~x[/tex]

    In that case i thought the answer was cot x. But that is wrong.

    Do you need to use integration by parts on this one?
     
    Last edited: Jan 7, 2010
  2. jcsd
  3. Jan 7, 2010 #2
    I just tried to evaluate this integral in Mathematica, but this does not return an analytical solution... so I doubt there is one. But maybe someone else has an idea?
     
  4. Jan 7, 2010 #3
    The integral sounds so unco and tricky at the first glance. But later you'll find out that it is equivalent to [tex]\int sin~x\cdot ln~x\ dx[/tex] after cancelling the cos(x) with that of tangant function. So letting f'(x) = sin(x) and g(x) = ln(x) and using integration by parts gives us [tex]-\cos \left( x \right) \ln \left( x \right) +{\it Ci} \left( x\right) [/tex]. Note that here we assume that x is greater than zero. otherwise the above antiderivative will get some complex term. At x=0, it is not defined.

    Here Ci(x) is some kind of special funcion called Cosine Integral which is exactly the second part in the integration you are supposed to do.

    AB
     
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