Answer:
The antiderivative is [tex]F(X) = 6x + 6x^4 + 3x^6 - 15[/tex].
Step-by-step explanation:
Antiderivative F(x)
This is the integral of [tex]F^{\prime}(x)[/tex]
So
F′(x) = f(x) = 6 + 24x^3 + 18x^5
Then:
[tex]F(x) = \int (6 + 24x^3 + 18x^5) dx[/tex]
[tex]F(x) = 6x + \frac{24x^4}{4} + \frac{18x^6}{6} + K[/tex]
[tex]F(x) = 6x + 6x^4 + 3x^6 + K[/tex]
F(1)=0
[tex]F(X) = 0[/tex] when [tex]x = 1[/tex]. We use this to find K.
[tex]F(x) = 6x + 6x^4 + 3x^6 + K[/tex]
[tex]0 = 6 + 6 + 3 + K[/tex]
[tex]K = -15[/tex]
Thus
The antiderivative is [tex]F(X) = 6x + 6x^4 + 3x^6 - 15[/tex].
