**1. Derivatives Properties Review.** Let be given. We recall the derivative

(1)

is defined by

(2)

We have several rules for differentiation:

**Power Rule:** If , then ;

**Product Rule:** We have ;

**Quotient Rule:** We have ;

**Composition Rule:** .

We want to introduce an “inverse” of differentiation. We will use the notation

(3)

interchangeably.

**2. Antidifferentiation.** A function is an **“Antiderivative”** of on an interval (usually it’s an open interval) iff

(4)

**Example 1.** Let . What is its antiderivative?

Well, we see its of the form

(5)

We take its derivative

(6)

Set to be equal to

(7)

and solve for

(8)

Thus

(9)

is the antiderivative for .

Note that is also an antiderivative for , where is any constant.

**Example 2.** Consider

(10)

What is its antiderivative?

We recall

(11)

which implies

(12)

for some constant , is the antiderivative.

**Example 3.** Take

(13)

and find its antiderivative.

We see that

(14)

We also remember that

(15)

which means

(16)

This gives us the antiderivative

(17)

where is some constant.

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