Answer:
The expression for the function is F(x) = (2/3)x^3-(3/2)x^6+4/3.
Step-by-step explanation:
If we use indefinite integration we can find the family of antiderivatives of f(x). This means that
[tex]F(x) = \int f(x)dx[/tex]
is an antiderivative of f(x). The, using the properties of the integral:
[tex]\int f(x)dx = 2\int x^2dx-9\int x^5dx = 2\frac{x^3}{3}-9\frac{x^6}{6} +C = \frac{2}{3}x^3 - \frac{3}{2}x^6 +C .[/tex]
Here, C stands for a generic real constant. We use the data F(1)=0 in order to find the exact value of C. Notice that
[tex]F(1) = \fra{2}{3}-\frac{3}{2}+C=-\frac{4}{3}+C=0.[/tex]
Then, [tex]C=\frac{4}{3}[/tex] and
[tex]F(x) = \frac{2}{3}x^3 - \frac{3}{2}x^6 +\frac{4}{3}.[/tex]
