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Math Help - u substituion in integration

  1. #1
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    u substituion in integration

    I cannot wrap my head around this u substitution business, please help.

    For example...

    \ln(\sin{x})=\int\ \frac{1}{\sin{x}}du

    (u = \sin{x})

    The integral is taken with respect to u, not x, which is totally worthless to me. I could care less about u. I'm all about x. So how do I get the integral to be dx ?

    other examples causing me trouble:

    \ln(e^{ix})=\int\ \frac{1}{e^{ix}}du (I need to show how this results in ix)

    or

    \ln(\sqrt{1-x^2})=\int\ \frac{1}{\sqrt{1-x^2}}du


    Thanks
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  2. #2
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    Quote Originally Posted by rainer View Post
    I cannot wrap my head around this u substitution business, please help.

    For example...

    \ln(\sin{x})=\int\ \frac{1}{\sin{x}}du

    (u = \sin{x})

    The integral is taken with respect to u, not x, which is totally worthless to me. I could care less about u. I'm all about x. So how do I get the integral to be dx ?

    Thanks
    ln(sin(x))=\int\frac{1}{sin(x)}du

    You have to find du in terms of x

    u=sin(x)\Rightarrow du=cos(x)dx

    \int\frac{du}{sin(x)}=\int\frac{cos(x)}{sin(x)}dx

    So the integral is:

    ln(sin(x))=\int cot(x)dx
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  3. #3
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    ln(e^{ix})=\int\frac{1}{e^{ix}}du

    u=e^{ix}\Rightarrow du=ie^{ix}dx

    \int\frac{1}{e^{ix}}du=\int\frac{ie^{ix}dx}{e^{ix}  }

    =\int i dx

    =ix
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