# What is the antiderivative of the function (x-3)*e^x?

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We have to find the integral of (x - 3)*e^x.

Here use integration by parts and the formula Int[ u dv] = u*v - Int[ v du]

u = (x - 3)

du = dx

v = e^x

dv = e^x dx

Int[(x - 3)*e^x] = (x - 3)*e^x - Int [e^x*dx]

=> (x - 3)*e^x - e^x

=> (x - 4)*e^x + C

**The required antiderivative of (x-3)*e^x = (x - 4)*e^x + C**

We'll integrate by parts. For this reason, we'll consider the formula:

Int udv = u*v - Int vdu (*)

We'll put u = x - 3 (1)

We'll differentiate both sides:

du = dx (2)

We'll put dv = e^x (3)

We'll integrate both sides:

Int dv = Int e^x dx

v = e^x (4)

We'll substitute (1) , (2) , (3) and (4) in (*):

Int udv = (x-3)*e^x - Int (e^x)dx

Int (x-3)*e^x dx = (x-3)*e^x - e^x + C

Int (x-3)*(e^x)dx = (e^x)*(x-3-1) + C

**The antiderivative of the given function is: Int (x-3)*(e^x)dx = (e^x)*(x-4) + C**