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Math Help - Integral4

  1. #1
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    Integral4

    \int \frac{dx}{x^4\sqrt{1+x^2}}



    answer:
    \frac{(2x^2-1)(1+x^2)^\frac{1}{2}}{3x^3}
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  2. #2
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    There are many ways, but I like trig sub for some reason.

    Let x=tan(u), \;\ dx=sec^{2}(u)du

    When we make the subs we get:

    \int csc(u)cot^{3}(u)du

    =\int csc(u)(csc^{2}(u)-1)cot(u)du

    Now, can you continue?. Another sub should do it, and when you get to the end remember to let u=tan^{-1}(x) to get it back in terms of x.
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  3. #3
    MHF Contributor kalagota's Avatar
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    also, from same substitution,

    \int \frac{\cos^3 u}{\sin^4 u} \, du

    t= \sin u
    dt = \cos u \, du
    \cos^2 u = 1-\sin^2 u = 1-t^2

    \int \frac{1-t^2}{t^4} \, dt
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  4. #4
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    Krizalid's Avatar
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    A reciprocal substitution also works.
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  5. #5
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    thanks
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  6. #6
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    Quote Originally Posted by Krizalid View Post
    A reciprocal substitution also works.
    I'ts very odd .... I'm certain I posted the same suggestion and some general details (it was post #2) but that post seems to have vanished.
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  7. #7
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    Dunno. I said it 'cause I saw nobody told it before.
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