Antiderivatives Examples

Let f(x) be a continuous function. Build an antiderivative F of f that satisfies F(2) = 0.

We know that if

then F(a) = 0. Since we want F(2) to be zero, we take a = 2 and define

Then

just like we wanted.

Let f(x) = cos (x²). Build an antiderivative G(x) of f(x) that satisfies G(2) = 7.

We just saw how to build an antiderivative F(x) of f(x) that satisfies F(2) = 0. This time we're given a specific function, so we put that in for the integrand:

We want a function that's 7 when x = 2, instead of being 0. Since we can add a constant to F(x) without changing its derivative, let

The derivative of G is

so G is still an antiderivative of f. And now

which is what we wanted.