Question

How do you find the antiderivative of `(4x^2 - 3x)e^x`?

Answer 1

`4e^x x^2-11e^x x+11e^x`

Explanation:

The antiderivative of a function is its integral. Here, we need to solve:

`int(4x^2-3x)e^xdx`

According to integration by parts, `intf(x)g(x)dx=f(x)intg(x)dx-intf'(x)(intg(x)dx)dx`.

Here, `f(x)=4x^2-3x` and `g(x)=e^x`.

But since `inte^xdx=e^x`, we can just write:

`e^x(4x^2-3x)-int(d/dx(4x^2-3x))e^xdx`

`e^x(4x^2-3x)-int(8x-3)e^xdx`

Integrating by parts the integral, we get:

`e^x(4x^2-3x)-e^x(8x-3)+int8e^xdx`

`e^x(4x^2-3x)-e^x(8x-3)+8e^x`

`e^x((4x^2-3x)-(8x-3)+8)`

`e^x(4x^2-3x-8x+3+8)`

`e^x(4x^2-11x+11)`

`4e^x x^2-11e^x x+11e^x`