Question

Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.)

f(x) =

2x4 + 4x3 − x
x3
, x > 0

F(x) =

2) Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.)

f ''(x) = 32x3 − 18x2 + 8x

f(x) =

Answer 1

1) F(x)=
`(2x^(5) )/(5)+x^(4)-(x^(2) )/(2)+C`

2)
`f(x)=(8x^5)/(5)-(3x^4)/(2)+(4x^3)/(3)+D`

Explanation:

Applying the antiderivation rules:

F(x)=
`(2x^(4+1) )/(4+1)+4(x^(3+1) )/(3+1)-(x^(1+1) )/(1+1)+C`

F(x)=
`(2x^(5) )/(5)+x^(4)-(x^(2) )/(2)+C`

Checking by differentitation we have:

`F'(x)=2*(5x^(4) )/(5)+4*(4x^3)/(4)+2*(x)/(2)\\F'(x)=2x^4+4x^3-x=f(x)`

Which is demonstrated.

2) To find f we must antiderivate twice:

`f'(x)=\int\limits {(32x^3-18x^2+8x)} \, dx\\f'(x)=8x^4-6x^3+4x^2+C\\f(x)=\int\limits( {8x^4-6x^3+4x^2)} \, dx\\ f(x)=8(x^5)/(5)-6(x^5)/(5)+4(x^3)/(3)+D\\ f(x)=(8x^5)/(5)-(3x^4)/(2)+(4x^3)/(3)+D`