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Math Help - Trying another integral of parts

  1. #1
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    Trying another integral of parts

    Integral sign with lower limit of 0 and upper limit of 1
    x^2e^2x dx
    So I found u = e^2x v=1/2(x)^3
    du=2e^2x dv= x^2 dx

    answer is 1/3(x^3e^2x)-1/12(x^4)(2e^2x) +c.
    NOT sure what to do with the limits.

    Do I have this correct above??
    thanks again
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  2. #2
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    Re: Trying another integral of parts

    You need to swap your u and v around. You'll end up in a loop if you continue to set u = e^{2x}

    u = x^2 \Leftrightarrow u' = 2x

    v' = e^{2x} \Leftrightarrow v = \dfrac{1}{2}e^{2x}

    \dfrac{1}{2}x^2e^{2x} - \int \dfrac{1}{2}e^{2x} \cdot 2x = \dfrac{1}{2}x^2e^{2x} - \int xe^{2x}

    Now do integration by parts again.
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    Re: Trying another integral of parts

    Ok looking at this, question. When I read online on how to find the deriviative of e^2x is was 2 in front, why half> Just asking, then what I read was wrong.
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    Re: Trying another integral of parts

    Quote Originally Posted by bradycat View Post
    Ok looking at this, question. When I read online on how to find the deriviative of e^2x is was 2 in front, why half> Just asking, then what I read was wrong.
    Yes, the chain rule: If u= 2x, then f(x)= e^{2x}= f(u)= e^u
    \frac{df}{dx}= \frac{df}{du}{\frac{du}{dx}= e^{u}(2)= 2e^{2x}

    But the integral goes the other way. To integrate \int e^{2x}dx, let u= 2x so that du= 2dx, dx= \frac{1}{2}du so the integral becomes \frac{1}{2}\int e^u du= \frac{1}{2}e^u+ C= \frac{1}{2}e^{2x}+ C.

    Differentiating e^{2x} requires, as you say, a multiplication by 2. Integration, being the inverse of differentiation, requires a division by 2.
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    Re: Trying another integral of parts

    Quote Originally Posted by bradycat View Post
    Integral sign with lower limit of 0 and upper limit of 1
    x^2e^2x dx
    So I found u = e^2x v=1/2(x)^3
    du=2e^2x dv= x^2 dx

    answer is 1/3(x^3e^2x)-1/12(x^4)(2e^2x) +c.
    NOT sure what to do with the limits.

    Do I have this correct above??
    thanks again

    \int x^2e^{2x}dx=Ax^2e^{2x}+Bxe^{2x}+Ce^{2x}+D


    Take derivative operation on both sides and find A,B,C and D.
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