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Math Help - Integration Problem (Fractions)

  1. #1
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    Integration Problem (Fractions)

    \displaystyle \int \frac{1}{x^2+12 x + 72}\,dx

    I don't know where to begin! Substitution definitely does not work. Integration by parts & partial fractions do not seem to work either.

    I cannot factor the denominator. I get imaginary roots!

    How do I proceed with this type of problem? Thanks. PS: I suspect it has something to do with the arctan function!

    Thanks!
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  2. #2
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    Krizalid's Avatar
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    substitution does actually work, the thing is that you need to do some algebra first.

    x^2+12x+72=(x+6)^2+36, so put x+6=6\tan t then the integral becomes \displaystyle\int{\frac{6{{\sec }^{2}}t}{36{{\tan }^{2}}t+36}\,dt}=\frac{1}{6}\int{dt}=\frac{1}{6}t+  k, now back substitute.
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  3. #3
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    Alternatively, to integrate \int \frac{1}{(x+ 6)^2+ 36} dx, let u= (x+ 6)/6 so that du= (1/6) dx, dx= 6du and you have
    \int \frac{1}{36u^3+ 36}(6du)= (1/6)\int \frac{1}{u^2+ 1}du

    which is (1/6)arctan(u)+ C and "back substitution" gives the same answer as Krizalid.
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  4. #4
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    Krizalid's Avatar
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    i was actually point that direction but when i pushed to button i didn't want to edit.

    actually, the latter approach is better than mine since introduces you a known integral that everyone junior should know.
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