Question

Consider the function f ( x ) = 2 x^2 − 9 x^5 . Let F ( x ) be the antiderivative of f ( x ) with F ( 1 ) = 0 . Then F ( x ).

Answer 1

The expression for the function is F(x) = (2/3)x^3-(3/2)x^6+4/3.

Explanation:

If we use indefinite integration we can find the family of antiderivatives of f(x). This means that

`F(x) = \int f(x)dx`

is an antiderivative of f(x). The, using the properties of the integral:

`\int f(x)dx = 2\int x^2dx-9\int x^5dx = 2(x^3)/(3)-9(x^6)/(6) +C = (2)/(3)x^3 - (3)/(2)x^6 +C .`

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

`F(1) = \fra{2}{3}-(3)/(2)+C=-(4)/(3)+C=0.`

Then,
`C=(4)/(3)` and

`F(x) = (2)/(3)x^3 - (3)/(2)x^6 +(4)/(3).`