TABLE OF CONTENTS
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
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.
Make it rain.