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Math Help - first order differential equation

  1. #1
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    first order differential equation

    can anyone show me how to integrate [(1+X)/(1-x)] dx ? im doing revision for my subject and came upon this question where im stuck.
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  2. #2
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    There various ways to go about it.

    \int\sqrt{\frac{1+x}{1-x}}dx

    Make the sub x=\frac{u^{2}-1}{u^{2}+1}, \;\ dx=\frac{4u}{(u^{2}+1)^{2}}du

    This whittles it down to:

    4\int\frac{1}{u^{2}+1}du-4\int\frac{1}{(u^{2}+1)^{2}}du

    Now, can you finish it up?.
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  3. #3
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    Quote Originally Posted by galactus View Post

    Make the sub x=\frac{u^{2}-1}{u^{2}+1}, \;\ dx=\frac{4u}{(u^{2}+1)^{2}}du
    im can solve it once you gave me that... but how do you know what to sub it with?
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  4. #4
    Behold, the power of SARDINES!
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    Quote Originally Posted by yen yen View Post
    can anyone show me how to integrate [(1+X)/(1-x)] dx ? im doing revision for my subject and came upon this question where im stuck.

    I would use an algebraic simplification first. Build the fraction using (1+x) to get

    \int \sqrt{\frac{1+x}{1-x}}dx=\int \sqrt{\frac{(1+x)^2}{1-x^2}}dx=\int\frac{1+x}{\sqrt{1-x^2}}dx=\int \frac{1}{\sqrt{1-x^2}}dx+\int \frac{x}{\sqrt{1-x^2}}dx

    The first is the arcsine and the 2nd is a u sub. I hope this helps.

    Note: I didn't use absolute values in the 3rd equality because the domain is x \in (-1,1) so the numerator is always posative there.
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