# What is the antiderivative of the function f(x) given by f(x)=x*e^8x ?

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The antiderivative of f(x)=x*e^8x can be found using integration by parts.

Int [u dv ]= u*v - Int[ v du]

Let u = x => du = 1

dv = e^8x => v = e^8x/8

Int [ x*e^8x ] = (x*e^8x)/8 - Int [1*(e^8x)/8]

=> (x*e^8x)/8 - (e^8x)/64

**The anti-derivative is (x*e^8x)/8 - (e^8x)/64 + C**

We'll integrate by parts, according to the identity below:

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

We'll put u = x. (1)

We'll differentiate both sides:

du = dx (2)

We'll put dv = e^8x (3)

We'll integrate both sides:

Int dv = Int e^8x dx

v = e^8x/8 (4)

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

Int udv = x*e^8x/8 - Int (e^8x/8)dx

**Int (x*e^8x)dx = x*e^8x/8 - e^8x/64 + C**