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Math Help - Integration: arctan(x) rule derivation

  1. #1
    Junior Member Greg98's Avatar
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    Integration: arctan(x) rule derivation

    Hello,
    the task is to derive integration rule (2) from rule (1).

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

    (2)  \int \frac{1}{x^2+c^2} dx = \frac{1}{c} \arctan (\frac{x}{c})

    c is constant and >0

    I tried to massage rule (1) with basic operations, but I always ended up with too many x's and c's. Also, I couldn't find any help using forum's search function. So, any help is welcome. Mucho thanks in advance!
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    let cu=x  \implies cdu = dx

    so your integral becomes

    \displaystyle \int \frac{dx}{x^2+c}=\int \frac{c du }{c^2u^2+c^2}=\frac{1}{c}\int \frac{du}{u^2+1}

    and the rest follows.
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