Integration: Antiderivatives and Indefinite Integrals =𝑥3 2𝑥=𝑥3−5 3𝑥=𝑥3+27 In...
Transcript of Integration: Antiderivatives and Indefinite Integrals =𝑥3 2𝑥=𝑥3−5 3𝑥=𝑥3+27 In...
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
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)