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

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Answer 1 · 2018-03-14T15:07:37

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

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:

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

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

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

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

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

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

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