How do I find the integral ∫(x⋅e−x)dx ?

1 Answer
Jacob F.
Aug 4, 2014

∫xe−xdx=−xe−x−e−x+C

Process:

∫xe−xdx= ?

This integral will require integration by parts. Keep in mind the formula:

∫udv=uv−∫vdu

We will let u=x, and dv=e−xdx.

Therefore, du=dx. Finding v will require a u-substitution; I will use the letter q instead of u since we are already using u in the integration by parts formula.

v=∫e−xdx
let q=−x.

thus, dq=−dx

We will rewrite the integral, adding two negatives to accommodate dq:

v=−∫−e−xdx

Written in terms of q:

v=−∫eqdq

Therefore,

v=−eq

Substituting back for q gives us:

v=−e−x

Now, looking back at the IBP's formula, we have everything we need to start substituting:

∫xe−xdx=x⋅(−e−x)−∫−e−xdx

Simplify, canceling the two negatives:

∫xe−xdx=−xe−x+∫e−xdx

That second integral should be easy to solve - it's equal to v, which we've already found. Simply substitute, but remember to add the constant of integration:

∫xe−xdx=−xe−x−e−x+C

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