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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