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Math Help - substitution technique

  1. #1
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    substitution technique

    The question i am given is:
    Use a substitution technique and then the table of integration to integrate
    ∫ x^5 √(x^4 4) dx
    Hint: x^5 = (x^3)(x^2)


    Tips on how to start this solution would be appreciated.

    Thanks
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  2. #2
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    let u = x^4 - 4 and du = 4x^3 dx

    x^5 can be split up into (x^3)(x^2) so when you make the substitutions, you integral will become: (1/4) integral of x^2 / sqrt(u) du. now to get x^2 in terms of u, take our original substitution u = x^4 - 4 and solve for x^2.

    x^4 = u + 4 so x^2 = sqrt(u+4) so the integral will become: (1/4)integral of sqrt[(u+4)/(u)] du. the integrand can be further simplified to sqrt(1 + 4/x) = sqrt(1 + (2/sqrt(x))^2). now consult your integral table and find the form that matches this and finish up.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by mellowdano View Post
    The question i am given is:
    Use a substitution technique and then the table of integration to integrate
    ∫ x^5 √(x^4 – 4) dx
    Hint: x^5 = (x^3)(x^2)


    Tips on how to start this solution would be appreciated.

    Thanks
    Start with the obvious substitution: u=x^4-4 so:

    I=\int x^5 \sqrt{x^4-4}\;dx=\int x^2 \sqrt{u}\;du=\int \sqrt{u+4}\sqrt{u}\;du=\int \sqrt{u^2+4u}\;du ...

    CB
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  4. #4
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    whoops sorry i misread the question. but same idea.
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  5. #5
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    Would this be correct?
    ∫ x^5 √(x^4 4) dx
    = ∫ ((x^2*x^2*x √(x^4 4)) dx


    Let u = x^2, then du/2 = xdx
    = ∫ 1/2 (u^2√u^2 4) du

    Let a = 4 and a^2 = 2
    = ∫(u^2√u^2 a^2) du
    = [x(u^2 a^2)^3/2 / 4] + [a^2x√(u^2 a^2) / 8] (a^4 /8) ln(x + √(u^2 a^2))
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