Integration: Antiderivatives and Indefinite Integrals =𝑥3 2𝑥=𝑥3−5 3𝑥=𝑥3+27 In this example, 1, 2 and 3 are all valid antiderivatives of (𝑥). ... Indefinite Integral

INTEGRATION: ANTIDERIVATIVES

AND INDEFINITE INTEGRALSMR. VELAZQUEZ

AP CALCULUS

INTRO TO INTEGRATION

Mathematicians spent a lot of time working with the topic of derivatives, describing how functions change at any given instant.

They then sought a way to describe how those changes accumulate over time, leading them to discover the calculation for area under a curve. This is known as integration, the second main branch of calculus.

Finally, Liebniz and Newton discovered the connection between differentiation and integration, known as the Fundamental Theorem of Calculus, an incredible contribution to the understanding of mathematics.

ANTIDERIVATIVES

Before we look at the various uses and techniques of calculating area, we will first examine the concept of the antiderivative.

A function 𝐹 is an antiderivative of 𝑓 on an interval 𝐼 if 𝐹′ 𝑥 = 𝑓(𝑥) for all 𝑥 in 𝐼.

Note that we call this function an antiderivative, and not the antiderivative.

It’s easy to see why this is true. Consider the function below:

𝑓 𝑥 = 3𝑥2

𝐹1 𝑥 = 𝑥3 𝐹2 𝑥 = 𝑥3 − 5 𝐹3 𝑥 = 𝑥3 + 27

In this example, 𝐹1, 𝐹2 and 𝐹3 are all valid antiderivatives of 𝑓(𝑥).

ANTIDERIVATIVES

If 𝐹 is an antiderivative of 𝑓 on a given interval 𝐼, then 𝐺 is also an antiderivative of 𝑓 if and only if 𝐺 is of the form 𝐺 𝑥 = 𝐹 𝑥 + 𝐶 for all 𝑥 in 𝐼, where 𝐶 is a constant.

Function Antiderivative

𝑔 𝑥 = 2𝑥 𝐺 𝑥 = 𝑥2 + 𝐶

ℎ 𝑥 =1

𝑥2𝐻 𝑥 = −

1

𝑥+ 𝐶

𝑗 𝑥 = cos 𝑥 𝐽 𝑥 = sin 𝑥 + 𝐶

NOTATION FOR ANTIDERIVATIVES

න𝑓(𝑥) 𝑑𝑥 = 𝐹 𝑥 + 𝐶

Indefinite Integral Antiderivative

IntegrandVariable of Integration

An antiderivative of 𝑓(𝑥)

Constant

It’s important to note that both of these terms—indefinite

integral and antiderivative—refer to the same thing.

BASIC INTEGRATION RULES

න0𝑑𝑥 = 𝐶

න𝑘 𝑑𝑥 = 𝑘𝑥 + 𝐶

න𝑘 𝑓(𝑥) 𝑑𝑥 = 𝑘න𝑓 𝑥 𝑑𝑥

න 𝑓 𝑥 ± 𝑔 𝑥 𝑑𝑥 = න𝑓 𝑥 𝑑𝑥 ±න𝑔 𝑥 𝑑𝑥

න𝑥𝑛 𝑑𝑥 =𝑥𝑛+1

𝑛 + 1+ 𝐶, 𝑛 ≠ −1

Integral of a Zero

Integral of a Constant

Constant Multiple Rule

Sum & Difference

Power Rule

BASIC INTEGRATION RULES

නcos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶

නsin 𝑥 𝑑𝑥 = −cos 𝑥 + 𝐶

නsec2 𝑥 𝑑𝑥 = tan 𝑥 + 𝐶

Trigonometric Integrals

නsec 𝑥 tan 𝑥 𝑑𝑥 = sec 𝑥 + 𝐶

නcsc 𝑥 cot 𝑥 𝑑𝑥 = −c𝑠𝑐 𝑥 + 𝐶

නcsc2 𝑥 𝑑𝑥 = cot 𝑥 + 𝐶

EXAMPLES

Find the indefinite integrals shown below, using basic integration rules.

න4𝑥 𝑑𝑥 න(cos 𝑥 + 3 sec2 𝑥)𝑑𝑥

EXAMPLES

Find the indefinite integrals shown below, using basic integration rules.

න 3𝑥4 − 5𝑥2 + 𝑥 𝑑𝑥

EXAMPLES

Find the indefinite integrals shown below, using basic integration rules.

න𝑥 + 1

𝑥𝑑𝑥

Hint: Rewrite the

integrand as two

fractions.

EXAMPLES

Find the indefinite integrals shown below, using basic integration rules.

නsin 𝑥

cos2 𝑥𝑑𝑥

Hint: Rewrite the

integrand as a product.

USING INITIAL CONDITIONS

We have seen that when we take an indefinite integral, we are actually getting the set of all possible antiderivatives of a function.

But suppose we want to find one particular antiderivative solution. For this, we would need an initial condition—in most cases this is simply the value of 𝐹 𝑥 for one specific value of 𝑥.

𝐹 𝑥 = න 2𝑥 − 1 𝑑𝑥 = 𝑥2 − 𝑥 + 𝐶

Suppose we know that the antiderivative passes through the point (2, 4). We can use this to find the particular solution.

GENERAL SOLUTION

𝐹 2 = 4 → 2 2 − 2 + 𝐶 = 4 → 4 − 2 + 𝐶 = 4 → 𝑪 = 𝟐

𝑭 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐 PARTICULAR SOLUTION

EXAMPLE

Find the antiderivative of 𝐹′ 𝑥 =1

𝑥2for 𝑥 > 0 which satisfies the condition

𝐹 1 = 0.

CLASSWORK & HOMEWORK

MATH JOURNAL: Don’t forget!

CLASSWORK: INDEFINITE INTEGRALS: Find the following indefinite

integrals. For the last two, find the particular solution using the initial

condition given.

න 12 + 𝑥 𝑑𝑥

න 𝑥 +1

2 𝑥𝑑𝑥

න𝑥2 + 2𝑥 − 3

𝑥4𝑑𝑥

𝐹 𝑥 = න 10𝑥 − 12𝑥3 𝑑𝑥 , 𝐹 3 = 2

𝐺 𝑥 = න2 sin 𝑥 𝑑𝑥 , 𝐺 0 = 1

1.

2.

3.

4.

5.

Homework:

Pg. 255-256, #1-63 (odd)