Question

Find an antiderivative F(x) with F′(x) = f(x) = 6 + 24x^3 + 18x^5 and F(1)=0.

Answer 1

The antiderivative is
`F(X) = 6x + 6x^4 + 3x^6 - 15`.

Explanation:

Antiderivative F(x)

This is the integral of
`F^(\prime)(x)`

So

F′(x) = f(x) = 6 + 24x^3 + 18x^5

Then:

`F(x) = \int (6 + 24x^3 + 18x^5) dx`

`F(x) = 6x + (24x^4)/(4) + (18x^6)/(6) + K`

`F(x) = 6x + 6x^4 + 3x^6 + K`

F(1)=0

`F(X) = 0` when
`x = 1`. We use this to find K.

`F(x) = 6x + 6x^4 + 3x^6 + K`

`0 = 6 + 6 + 3 + K`

`K = -15`

Thus

The antiderivative is
`F(X) = 6x + 6x^4 + 3x^6 - 15`.