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Thread: Integration, x^3*sqrt(2-x)

  1. #1
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    Unhappy Integration, x^3*sqrt(2-x)

    hello, this problem has been annoying me for the past 24 hours, and i cant seem to figure out where to begin,

    Evaluate the integral:
    (integration sign goes here) (x^3*sqrt(2-x)) dx

    ive tried substitution but got nowhere, i attempted integ. by parts, but got confused because my teacher never taught it.

    this question is on my final exam review, and it says the answer is:
    -(( 16/3) * (2-x)^(3/2) - (24/5) * (2-x)^(5/3) + (12/7) * (2-x)^(7/2) - (2/9)*(2-x)^(9/2)) + C

    thank you.
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  2. #2
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    Quote Originally Posted by micro31337 View Post
    hello, this problem has been annoying me for the past 24 hours, and i cant seem to figure out where to begin,

    Evaluate the integral:
    (integration sign goes here) (x^3*sqrt(2-x)) dx

    ive tried substitution but got nowhere, i attempted integ. by parts, but got confused because my teacher never taught it.

    this question is on my final exam review, and it says the answer is:
    -(( 16/3) * (2-x)^(3/2) - (24/5) * (2-x)^(5/3) + (12/7) * (2-x)^(7/2) - (2/9)*(2-x)^(9/2)) + C

    thank you.
    Let u = 2-x, expand and integrate.
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  3. #3
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    so u mean do substitution?
    so
    u = 2-x, x = 2 - u
    du = -1dx
    -du = dx

    substituting in u: -(ingetral symbol)((2-u)^3 * u^(1/2))du
    is that what you mean? i dont see how i can continue with this
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  4. #4
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    Quote Originally Posted by micro31337 View Post
    so u mean do substitution?
    so
    u = 2-x, x = 2 - u
    du = -1dx
    -du = dx

    substituting in u: -(ingetral symbol)((2-u)^3 * u^(1/2))du
    is that what you mean? i dont see how i can continue with this
    So -\int (2-u)^3 \sqrt{u} du = -\int \left(8 - 12u + 6u^2 - u^3\right) u^{1/2}\,du = \int u^{7/2} - 6 u^{5/2} + 12 u^{3/2} - 8 u^{1/2}\,du.

    Then integrate term by term.
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  5. #5
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    Ooh wow, i had forgotten i could do that lol, thank you man!
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