17Calculus - Integrals and Integration
17Calculus
Integrals and Integration
The idea of integration is that integration 'undoes' the derivative, i.e. integration is the 'inverse' of differentiation. I used quotes around the word 'inverse' because in actuality differentiation and integration are mostly inverses but not completely. Let's start with what an integral is and some notation.
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If you have already studied integration and just want some practice problems, there are several places that you can find lots of practice problems on our site. This page contains basic practice problems for first semester calculus for students who are just starting to learn integration.
If you are in first semester calculus and you have learned all or most of the techniques you need for the course, our Calculus 1 Practice Integrals page will help you prepare for your integrals exam and your final exam. Those problems are mixed just as you will find on your exams.
If you are in second semester calculus and you have learned all or most of the advanced integrals, our Calculus 2 Practice Integrals page will help you prepare for your integrals exam and your final exam. Those problems are mixed just as you will find on your exams.
What Is An Integral?
Before we get to the details of integration, let's watch a quick video on exactly what integrals are. The material in this video is covered on several 17calculus pages but the video gives you a good overview of integration, what it means and some of the notation.
Physical Chemistry - What is an integral? [7min-21secs]
video by Physical Chemistry |
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As you know, if you have a function, \(g(x)\), the derivative of \(g(x)\) is written
\(\displaystyle{
\frac{d}{dx}[g(x)] = G(x)
}\)
In this equation, we are using \(G(x)\) to represent the new function we get after taking the derivative. When integrating, the notation looks like this
\(\int{F(x)~dx} = f(x) + C \)
In this equation, \(F(x)\) is the function we are integrating and \(f(x)+C\) is the result. The curved vertical line \(\int{}\) and the \(dx\) are both necessary in this notation. They act like brackets to indicate what is being integrated, which we call the integrand.
Okay, so what are integrals and how do we go about calculating them? Here is a great introduction video to integrals, what they represent and the notation. This is one of the best introductory videos you will find anywhere on any topic. So your time will be well spent watching this.
Krista King Math - Integrals - Calculus [ excellent video ] [5min-55secs]
video by Krista King Math |
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You will notice in the equation \(f(x)+C\), we have \(+C\) at the end. We need this because the derivative of a constant is zero. When we go in reverse, it is not possible to recover constants without more information. So, we use the \(+C\) as a placeholder for an unknown constant. Here is a video that explains this very well. If you are already confident with integrals, you can easily skip this video without losing anything. But, if you need a bit more explanation with some examples, this video may help.
Khan Academy - The Indefinite Integral or Anti-derivative [9min-28secs]
In this video he starts with the derivative \( \frac{d}{dx}[x^2] = 2x \).
Then explains how to work this backwards, i.e. given \(2x\) how do you get \(x^2\). Then he shows that the derivative of \( x^2+1\) is also \(2x\) and the derivative of \( x^2+2 \) is also \(2x\). So that the derivative of \( x^2 + any~constant\) is \( 2x\), usually written \(x^2+C\) where \(C\) just represents a constant.
Then he looks at the derivative of \(y=Ax^n\) which is \( (A \cdot n) x^{n-1} \) to try to figure out what \( y=\int{x^3 ~ dx} \) is.
This is a neat way of learning integration. He then generalizes the power rule for integration from this example.
Another example: \( \int{5x^7 dx} \) He is correct about not forgetting the \(+C\). Many instructors will take off points if you leave it off.
To sum up, although his explanation is a little choppy, this is a pretty good introduction video to integration.
video by Khan Academy |
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Some Basic Formulas
Here are a few basic formulas that you will need for upcoming pages. You already know the derivatives of these, so the anti-derivatives should contain no surprises.
\(\displaystyle{ \int{ k~dx } = kx + C }\) |
\(\displaystyle{ \int{ x^r~dx } = \frac{x^{r+1}}{r+1} + C }\), \(r \neq -1\) | |
\(\displaystyle{ \int{ e^x~dx } = e^x + C }\) |
\(\displaystyle{ \int{ \frac{1}{x} dx } = \ln|x| + C }\) | |
\(\displaystyle{ \int{ \sin(x) ~dx} = -\cos(x) + C }\) |
\(\displaystyle{ \int{ \cos(x) ~dx } = \sin(x) + C }\) |
Notes
1. In the above list, \(k\) is a constant and \(r\) is a rational number.
2. For the natural logarithm, be careful that you understand that \( \int{ \ln(x)~ dx } \neq 1/x + C \). This is a common mistake when students are first learning integration.
3. Watch the negative on the integral for sine.
4. There are comparable anti-derivatives for the other trig functions as well. You can find those details on the trig integration page.
More Discussion on \(\displaystyle{ \int{ \frac{1}{x} dx } = \ln|x| + C }\)
As we say on the learning tools page, what you learn in class and from any textbook is just a glimpse of calculus. Your instructor cannot present EVERYTHING about calculus or any subject. This video will give you an idea of what this looks like. We highly recommend that you watch it. He uses graphs to explain really well why \(\displaystyle{ \int{ \frac{1}{x} dx } = \ln|x| + C }\) is not the whole picture.
Trefor Bazett - Your calculus prof lied to you (probably)
video by Trefor Bazett |
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Okay, time for some practice problems.
Practice
For additional practice integrals, see the Calculus 1 Practice Integrals page.
Unless otherwise instructed, evaluate these integrals, giving your answers in exact form.
\( \int{ x^{2/3} ~dx } \)
Problem Statement
Evaluate the integral \( \int{ x^{2/3} ~dx } \). Give your answer in simplified, factored form.
Solution
Integrals ForYou - 118 video solution
Comment On Notation - Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.
video by Integrals ForYou |
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\(\displaystyle{\int{3x^2+2x+1~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{3x^2+2x+1~dx}}\), giving your answer in exact form.
Solution
Krista King Math - 996 video solution
video by Krista King Math |
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\(\displaystyle{\int{3x^4+5x-6~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{3x^4+5x-6~dx}}\), giving your answer in exact form.
Solution
Krista King Math - 997 video solution
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\(\displaystyle{\int{1-2x^2+3x^3~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{1-2x^2+3x^3~dx}}\), giving your answer in exact form.
Solution
Krista King Math - 998 video solution
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\(\displaystyle{\int{\frac{-1}{x^2}dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{\frac{-1}{x^2}dx}}\), giving your answer in exact form.
Solution
Krista King Math - 999 video solution
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\(\displaystyle{\int{\frac{3}{x^3}+2x^{3/2}-1~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{\frac{3}{x^3}+2x^{3/2}-1~dx}}\), giving your answer in exact form.
Solution
Krista King Math - 1000 video solution
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\(\displaystyle{\int{x^{5/2}-\frac{5}{x^4}-\sqrt{x}~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{x^{5/2}-\frac{5}{x^4}-\sqrt{x}~dx}}\), giving your answer in exact form.
Solution
Krista King Math - 1001 video solution
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\(\displaystyle{\int{\frac{3}{2}x^{1/2}+7~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{\frac{3}{2}x^{1/2}+7~dx}}\), giving your answer in exact form.
Solution
Krista King Math - 1002 video solution
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\(\displaystyle{\int{1+2x-4x^3~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{1+2x-4x^3~dx}}\), giving your answer in exact form.
Solution
PatrickJMT - 1005 video solution
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\(\displaystyle{\int{(x+1)(x^2+3)~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{(x+1)(x^2+3)~dx}}\), giving your answer in exact form.
Solution
PatrickJMT - 1007 video solution
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\(\displaystyle{\int{-5e^x+7\sin(x)~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{-5e^x+7\sin(x)~dx}}\), giving your answer in exact form.
Solution
PatrickJMT - 1315 video solution
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\(\displaystyle{\int{2e^x-1+\sin(x)\csc(x)~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{2e^x-1+\sin(x)\csc(x)~dx}}\), giving your answer in exact form.
Solution
PatrickJMT - 1316 video solution
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\(\displaystyle{\int{\frac{2}{x^{3/4}}-\frac{3}{x^{2/3}}~dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{\frac{2}{x^{3/4}}-\frac{3}{x^{2/3}}~dx}}\), giving your answer in exact form.
Solution
Krista King Math - 1003 video solution
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\(\displaystyle{\int{2x\sqrt{x}-\frac{1}{\sqrt{x}}dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{2x\sqrt{x}-\frac{1}{\sqrt{x}}dx}}\), giving your answer in exact form.
Solution
Krista King Math - 1004 video solution
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\(\displaystyle{\int{\frac{3x-2}{\sqrt{x}}dx}}\)
Problem Statement
Evaluate \(\displaystyle{\int{\frac{3x-2}{\sqrt{x}}dx}}\), giving your answer in exact form.
Solution
PatrickJMT - 1006 video solution
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Practice Instructions
Unless otherwise instructed, evaluate these integrals, giving your answers in exact form.
