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Math Help - Substitutions & Partial Fractions

  1. #1
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    Substitutions & Partial Fractions

    \int \frac {\sqrt{4-x^2}}{x}dx

    I was thinking along the lines of x = \cos u because \sin^2x = 1 - \cos^2 x, but I don't know where to take it.


    Also, a partial fractions problem:

    \int \frac {1}{x(x+1)}dx

    These things always turn out to be much simpler than they look.
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  2. #2
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    try using x=2cos(u). then you can separate the 4 in the numerator i.e 4-x^2=4(1-cos(u)^2)=4sin(u)^2 which is easy to find the square root of. the rest is pretty straightforward.

    The second partial fraction seems relatively simple, all you need to do is let 1/x(x+1)=(A/x)+ (B/(x+1)), then multiplying through by x and then x+1 gives

    1=A(x+1) + B(x)

    then solve by substituting appropriate values for x, i.e let x=0 and then let x=-1 to find A and B respectively. you will then end up with two separate fractions to integrate with only one function of x in each denominator.
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