Integration and Antiderivatives
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Objectives
- Learn what antiderivatives are and their relation ship to functions
- Understand the relationship between antiderivatives and derivatives and why they are very very nearly (but not quite) inverse operations
- Similar to derivatives, know the linear operator rules for working with sums, differences, and constant coefficients
- Learn what the process of indefinite integration means
One of the tools we will need to find specific areas is the process of finding a function's antiderivative. This first lesson will conceptually familiarize us with what they are and what we need to know about them.
Practice problems and worksheet coming soon!
Going the Other Way
Antiderivatives, often referred to as integrals, are the other half of Calculus. In many ways, derivatives and integrals are like multiplication and division in that they complete one another.Simply stated, antiderivatives are about starting a problem with a derivative and looking for what original expression you would have had.A simple example: an antiderivative of $\cos(x)$ is $\sin(x)$, because the derivative of $\sin(x)$ is $\cos(x)$.Vocab FYI
We refer to the process of finding antiderivatives as integration the same way that we referred to the process of finding derivatives as differentiation.What To Expect
One of the main differences you'll find with antiderivatives is that it will seem more like an art, where derivatives seem more like a technique. One reason for this is that you can actually find the derivative of any one-variable function you could come up with, but the same cannot be said for antiderivatives.Keep this in mind as you learn. Ultimately, you'll develop strong intuition for writing integrals, as well as a short list of techniques to try (and in what order to try them to make your life easiest).The next several lessons will get you started on integrating specific functions, but see if you can"t impress yourself with the following problems by using what you already know about derivatives.In the following two examples, find an expression that is an antiderivative to the given expression.Show solution
$\blacktriangleright$ Think about what you would have taken the derivative of to get $3x^2$: How about $x^3$? We can say that $x^3$ is an antiderivative of $3x^2$ for exactly this reason.
Show solution
$\blacktriangleright$ Since the derivative of $e^x$ is $e^x$, we can correctly say that $e^x$ is an antiderivative of $e^x$.
Integration Notation
To communicate the idea of finding an antiderivative, mathematicians use the integral symbol:$$\int$$Whenever you see this symbol in front of an expression, it is asking you to find the antiderivative of that expression. We'll continue to see and discuss this symbol as our studies continue. For now, you should know the following few facts solidly:- The $\int$ sign without any number on top or bottom of it is a symbol that tells you to take the antiderivative of the expression that follows it
- Following the integral symbol and the expression, you need to write $dx$ (assuming $x$ is the function variable)
- After every antiderivative you find, you need to include $+C$ to represent an unknown integration constant (more on this in a minute)
Warning!
Failing to write $dx$ or $+C$ as part of the antiderivative process will lose you points on a test. It's nit-picky but important.You Should Know
There will soon be a difference between integrals that do and do not have "limits" or "bounds" notated by having little numbers above and below the integral symbol. In this lesson we are talking exclusively about integrals without limits, often called indefinite integrals. The notation and conventions required, such as writing $dx$ after the integral and adding a $+C$ to your answer, are always needed for these integrals.Formal Definition
Using this common function notation, let's finally formally define the antiderivative of a function.Define: AntiderivatesLet $f(x)$ be a one-variable function of $x$. We say that $F(x)$ is the antiderivative of $f(x)$ if$$\frac{d}{dx} \, F(x) = f(x)$$
This is why the word antiderivative is used - it's literally defined by using derivatives backward.The Infamous + C
Back in Examples 1 and 2, I deliberately said "an antiderivative" instead of "the antiderivative", and for good reason. When we seek out antiderivatives, we need to understand that $F(x)$ may have had a constant added to it that we couldn't have known about. The simplest way to understand is to re-answer Example 1 with a slightly different question.TheoremIf $F(x)$ is an antiderivative of $f(x)$, then $F(x) + C$ is also an antiderivative of $f(x)$
For this reason, you must write +C at the end of every antiderivative you answer in this course, whether it is a test, graded homework or even ungraded homework. No lost points in Calc exams could be more preventable, or sadly, more common.Once more for the people in the back:Write a +C after every single antiderivative answer you give for the rest of eternity!!!Pro Tip
Go write the prior sentence down 20 times.Linear Operation
Just like derivatives are linear operators », so too are antiderivatives. Namely, this means two important but intuitive things: first that constants can "come outside" of antiderivatives, and second that the antiderivative of a sum of terms is the sum of the antiderivatives of each term.These two ideas will become completely automatic to you as you go through the course.For example, if$$\int \cos(x) \, dx = \sin(x) + C$$and$$\int e^x \, dx = e^x + C$$then$$\int \cos(x) + e^x \, dx = \int\cos(x) \, dx + \int e^x \, dx$$$$=\sin(x) + e^x + C$$(we only need one $+C$ for any integral)You usually aren't tested on this explicitly but there is one additional example below to reinforce this idea.Put It To The Test
We'll save the heavy stuff for the following lessons, once we dive into specific function patterns, but make sure you can answer the following questions for now.Show solution
$\blacktriangleright$ While there is of course no one correct answer for your own words, hopefully your definition has something about antiderivatives being the backward operation of derivatives. I like to think of a function's antiderivative as the thing you would take the derivative of to get the function.
Show solution
$\blacktriangleright$$$\int \ln(x) \, dx$$Make sure you include both the integral symbol and the $dx$ differential symbol.
Show solution
$\blacktriangleright$ The right question to ask is, what function's derivative is $\sec(x) \tan(x)$? That function will be the antiderivative of $\sec(x) \tan(x)$.Digging back to what we know about trig derivatives, the answer is $\sec(x) + C$ because$$\frac{d}{dx} \, \big[ \sec(x) + C \big] = \sec(x) \tan(x)$$In proper notation, we write$$\int \sec(x) \tan(x) \, dx = \sec(x) + C$$
Show solution
$\blacktriangleright$ The derivative of some $x^5$ polynomial term will be an $x^4$ term, so we just have to figure out the constant coefficient. Either with some reasoning or guess and check, it is $2x^5$ whose derivative is $10x^4$, and so:$$\int 10x^4 \, dx = 2x^5 + C$$
Show solution
$\blacktriangleright$ Typically, parent functions imply coefficients of $1$ and no sums. What we should write here is each term as its own function, knowing that the integral of a sum is the sum of each integral. We should also factor constants out in front.$$\int 3x^6 - 2e^x \, dx$$$$= 3 \int x^6 \, dx - 2\int e^x \, dx$$Notice that we need a $dx$ differential for each integral.
- Begin to understand the role of antiderivatives as a backward process from what you already learned about derivatives
- Remember to include $+C$ at the end of your antiderivative answers
- Learn the common notation associated with antiderivatives, including $F(x)$, $\int$, and remembering to write the $dx$ at the end of the antiderivative question
- Apply sum / difference and constant coefficient logic to integrals as linear operators, similar to how derivatives are linear operators
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