How do you find the antiderivative of ((2x)e^(3x))?

1 Answer
Aug 3, 2016

1/9*(3x-1)*e^(3x)+C.

Explanation:

Let, I=int2xe^(3x)dx rArr I=2intxe^(3x)dx.

To find I, we will use the following Rule of Integration by Parts :

intuvdx=uintvdx-int{(du)/dxintvdx}dx.

We take, u=x, so, (du)/dx=1, &, v=e^(3x), so, intvdx=1/3e^(3x). So,

I=x*1/3e^(3x)-int{1*1/3e^(3x)}dx

=x/3e^(3x)-1/3inte^(3x)dx

=x/3e^(3x)-1/3*1/3e^(3x)

:. I = 1/9*(3x-1)*e^(3x)+C.