Fluther: How do you integrate (x-x^2)e^x?

How do you integrate (x-x^2)e^x?

) June 5th, 2013
6 responses
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I’ve attempted integration by parts and I can’t seem to get the correct answer. Would you be able to show me through the steps?

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Topics: engineering, calculus, mathematics, antiderivatives

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Answers

Yep, you’ll need integration by parts! Two rounds of it. Lucky you :P

Start by choosing one term to be u and one term to be dv. You’ll have to take the derivative of u and integrate dv. The important thing is to always choose u to be something that gets simpler when you take the derivative. e^x remains e^x when you derive, so don’t choose that.

u = (x-x^2), dv = e^x
du = 1–2x, v = e^x

Now your formula is:

integral(udv) = uv – integral(vdu)
So integral((x-x^2)e^x) = (x-x^2)e^x – integral((1–2x)e^x)
But in order to solve that second integral, (1–2x)e^x, you have to use integration by parts again!

Choose u = 1–2x, dv = e^x
du = -2, v = e^x

integral((1–2x)e^x) = (1–2x)e^x – integral(-2e^x)

Okay, NOW the integral on the righthand side is easily solvable! In fact its integral is itself: -2e^x.

Plug THAT in to the first result from integration by parts:

integral((x-x^2)e^x) = (x-x^2)e^x – integral((1–2x)e^x)
= (x-x^2)e^x – ((1–2x)e^x – integral(-2e^x))
= (x-x^2)e^x – ((1–2x)e^x – (-2e^x))

Now just distribute your negative signs:
= (x-x^2)e^x – (1–2x)e^x – 2e^x
You could leave it there, or clean it up a bit by distributing your terms and combining like terms:
xe^x – x^2e^x – e^x + 2xe^x – 2e^x
-x^2e^x + 3xe^x -3e^x

Done!

Mariah (25883

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