Integration by parts

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integrate (e^x-x)^2 dx

  • Integration by parts -

    I would expand it first
    ∫ (e^x - x)^2 dx
    = ∫ e^(2x) - 2xe^x + x^2 dx

    let's concentrate on the middle term, since the other two terms are easy to integrate

    ∫ 2xe^x dx

    let u = 2x , let dv = e^x dx
    du = 2dx and v = e^x

    then ∫ 2xe^x dx = (2x)(e^x) - ∫ 2e^x dx
    = 2xe^x - 2e^x

    finally ...
    ∫ (e^x-x)^2 dx
    = ∫ e^(2x) - 2xe^x + x^2 dx
    = (1/2e^(2x) - (2xe^x - 2e^x) + (1/3)x^3 + c

    = (1/2)e^(2x) - 2xe^x + 2e^x + (1/3)x^3 + c

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