The FTC tells us how to build antiderivatives for any continuous function we like. No user manuel required. There are some things we need to know about these antiderivatives.

\item If we define an antiderivative of *f*(*x*) by

then

because we're integrating over an interval of length 0.

1) If we define two antiderivatives of *f*(*x*) by

where *a* < *b*, then the difference between *F*_{b(x)} and *F*_{a(x)} is

which is a constant.

2) Define an antiderivative of *f*(*x*) by

Since the derivative of a constant is 0, we can add any constant *c* that we like to *F*(*x*) and get a new antiderivative of *f*(*x*). That is, the function

has *G*'(*x*) = *f*(*x*).

Using the information above, we can build antiderivatives with various properties.

## Practice:

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 (*x*^{2}). 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. | |

Build an antiderivative *F*(*x*) of *f*(*x*) = *e*^{ex} satisfying *F*(8) = 0.

Answer

because

Build an antiderivative *F*(*x*) of *f*(*x*) = *e*^{ex} satisfying *F*(8) = -1.

Answer

We take the antiderivative from the first problem and add -1:

Build an antiderivative of cos *x* that is 0 when *x* = 9.

Answer

This problem is asking for a function *F*(*x*) that is an antiderivative of *f*(*x*) = cos *x* and satisfies *F*(9) = 0. We know how to do this:

Actually, in this case we could finish integrating and get a nice formula:

Build an antiderivative of sin (*x*^{2}) that is 3 when *x* = 8.

Answer

The function

is an antiderivative of sin(*x*^{2}) that is 0 when *x* = 8. Since we want an antiderivative that equals 3 when *x* = 8, we just add 3 to this integral: