is an * antiderivative* of if

Notation:

For example,

In fact, all of the following functions are antiderivatives of , because they all differentiate to :

This is the reason for the " " in the
notation: You can add any *constant* to the "basic"
antiderivative and come up with another
antiderivative.

C is called the * arbitrary constant*.

* Remark.* (a) Antiderivatives are often referred
to as * indefinite integrals*, and sometimes I'll
refer to as "the integral
of with respect to x". This terminology is actually
a bit misleading, but it's traditional, so I'll often use it. There
is another kind of "integral" --- the *
definite integral* --- which is probably more deserving of the
name.

(b) The notation " " will also be used for *
definite integrals*. The integral sign is a stretched-out "S", and comes
from the fact that definite integrals are defined in terms of * sums*.

" " is a
mathematical object called an * operator*, which
roughly speaking is a function which takes functions as inputs and
produces functions as outputs. Despite appearances, " " isn't a separate thing; in fact, " " is the
* whole name* of the antidervative operator. It's
a weird name --- it consists of three symbols (" ", "d", and "x"),
and has a space between the " "
and the " " for the input function.

I'll come back to this again when I discuss *
substitution*, since at that point this can become a source of
confusion.

Every differentiation formula has a corresponding antidifferentiation formula. This makes it easy to derive antidifferentiation rules from the rules for differentiation.

* Theorem.* (* Power Rule*)
For ,

* Proof.* This follows from the fact that

(Notice that the expression on the left is undefined if .)

* Example.* Compute the following
antiderivatives:

(a) .

(b) .

(c) .

(d) .

(a)

(b)

(c)

(d)

* Theorem.*

* Proof.* I'll prove the first formula by way of
example; see if you can prove the others.

Suppose that

By definition, this means that

By the rule for the derivative of a sum,

By definition, this means that

* Example.* Compute the following
antiderivatives:

(a) .

(b) .

(a)

(b)

Since the derivative of a product is *not* the product of the
derivatives, you can't expect that it would work that way for
antiderivatives, either.

* Example.* Compute .

To do this antiderivative, I *don't* antidifferentiate and separately. Instead, I multiply
out, then use the rules I discussed above.

Likewise, the derivative of a quotient is *not* the quotient
of the derivatives, and it doesn't work that way for antiderivatives.

* Example.* Compute .

Don't antidifferentiate and separately! Instead, divide the bottom into the top:

Every differentiation rule gives an antidifferentiation rule. So

* Example.* Compute .

For example,

* Example.* and . Find
y.

To find y, antidifferentiate :

:

Therefore,

This process is a simple example of * solving a
differential equation with an initial condition*.

* Example.* Suppose an object moves with constant
acceleration a. Its initial velocity is , and its initial position is . Find its position function .

First, , so

When , , so

Therefore,

Next, , so

When , :

Therefore,

For example, an object falling near the surface of the earth
experiences a constant acceleration of -32 feet per second per second
(negative, since the object's height s is *decreasing*). Its
height at time t is

Here is its initial velocity and is the height from which it's dropped.

Copyright 2018 by Bruce Ikenaga