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

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

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

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

$\int f(x)dx = 2\int x^2dx-9\int x^5dx = 2\frac{x^3}{3}-9\frac{x^6}{6} +C = \frac{2}{3}x^3 - \frac{3}{2}x^6 +C .$

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

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

Then, $C=\frac{4}{3}$ and

$F(x) = \frac{2}{3}x^3 - \frac{3}{2}x^6 +\frac{4}{3}.$