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Math Help - Integration by parts with a simple substitution

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    Red face Integration by parts with a simple substitution

    Integrate: (1/3)(x)[e^(x^1/3)]dx

    I have tried making u=1/3x and dv=e^(x^1/3)
    I've also tried to do a simple substitution first. I keep getting stuck and I've tried every combination I can think of. I would greatly appreciate any help. Thanks so much!
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    Quote Originally Posted by houndmom87 View Post
    Integrate: (1/3)(x)[e^(x^1/3)]dx

    I have tried making u=1/3x and dv=e^(x^1/3)
    I've also tried to do a simple substitution first. I keep getting stuck and I've tried every combination I can think of. I would greatly appreciate any help. Thanks so much!
    integrate (x/3)Exp[x^(1/3)&# 93; - Wolfram|Alpha (Click on show steps)
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    I guess I'm just unsure as to how to get u^5. Thanks!
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    Quote Originally Posted by houndmom87 View Post
    I guess I'm just unsure as to how to get u^5. Thanks!
    Have you tried doing the algebra? Please show your attempt with this calculation.
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    Quote Originally Posted by houndmom87 View Post
    Integrate: (1/3)(x)[e^(x^1/3)]dx

    I have tried making u=1/3x and dv=e^(x^1/3)
    I've also tried to do a simple substitution first. I keep getting stuck and I've tried every combination I can think of. I would greatly appreciate any help. Thanks so much!
    But it isn't the "1/3x" that is the problem! It is that "x^1/3" in the exponential. I recommend letting u= x^{1/3}. Then du= (1/3)x^{-2/3}dx so that dx= 3x^{2/3}du. That way (1/3)xdx= (1/3)x(3x^{2/3}du)= x^{5/3}du. And since u= x^(1/3), x^(5/3)= (x^(1/3))^5= u^5.

    That is, (1/3)xdx= u^5du and so (1/3)xe^{x^1/3}dx= u^5e^udu.

    Now that's still going to be a nuisance- use "integration by parts"- five times!
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