Question

Which of the following is an antiderivative of f(x)=x3e4x? Hint: Do not integrate directly. Differentiate and check.

Answer 1

If we do not integrate directly, we need to differentiate the answers and check, so let's start:

a)

`((1)/(16)x^4e^(4x)-1)^(\prime)=(1)/(16)(4x^3e^(4x)+x^4e^(4x)4)=e^(4x)x^3(4)/(16)(1+x)`

So we discard this option.

b)

`((1)/(128)e^(4x)(32x^3-24x^2+12x-3)+10)^(\prime)=(1)/(128)(4e^(4x)(32x^3-24x^2+12x-3)+e^(4x)(96x^2-48x+12))`

If we simply:

`(1)/(128)e^(4x)(128x^3-96x^2+48x-12+96x^2-48x+12)`

This is equal to:

`e^(4x)x^3`

Then the correct answer is B.