How do you integrate by parts (x⋅e4x)dx?

1 Answer
Stefan V.
Aug 29, 2015

∫(x⋅e4xdx)=116⋅e4x⋅(4x−1)+c

Explanation:

Remember that the formula that allows you to integrate a function by parts looks like this

∫(u⋅dv)=u⋅v−∫(v⋅du) , where

u, v - are two functions of x;
du, dv - their derivatives.

So, you need to identify u and dv, then calculate du and v.

If you take u=x and dv=e4x, you will have

u=x⇒du=dx

and

dv=e4x⇒v=∫(e4xdx)=14⋅e4x

Your target integral will thus be

∫(x⋅e4xdx)=x⋅14⋅e4x−∫(14e4x⋅dx)

∫(x⋅e4xdx)=14⋅x⋅e4x−14⋅14⋅e4x+c

∫(x⋅e4xdx)=14e4x(x−14)+c

∫(x⋅e4xdx)=116⋅e4x⋅(4x−1)+c

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