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Integrals |
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Integral FAQs
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] | |
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 [ best video ] | |
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 | |
Some Basic Formulas |
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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.
\(\displaystyle{ \int{ e^x~dx } = e^x + C }\) |
\(\displaystyle{ \int{ \frac{1}{x} dx } = \ln(x) + C }\) | |
\(\displaystyle{ \int{ \sin(x) ~dx} = -\cos(x) + C }\) |
\(\displaystyle{ \int{ \cos(x) ~dx } = \sin(x) + C }\) |
Notes
1. 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.
2. Watch the negative on the integral for sine.
3. There are comparable anti-derivatives for the other trig functions as well. You can find those details on the trig integration page.
faq: Why do integrals always have a dx? |
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Before you try some practice problems, here is a video that answers a very common question about integrals.
Krista King Math - Why do integrals always have a dx? [4min-36secs] | |
Okay, time for some practice problems. |
next: integration by substitution → |
Practice Problems |
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Instructions - - Unless otherwise instructed, evaluate the following integrals, giving your answers in exact form.
Level A - Basic |
Practice A01 | |
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\(\displaystyle{\int{3x^2+2x+1~dx}}\) | |
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solution |
Practice A02 | |
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\(\displaystyle{\int{3x^4+5x-6~dx}}\) | |
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solution |
Practice A03 | |
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\(\displaystyle{\int{1-2x^2+3x^3~dx}}\) | |
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solution |
Practice A04 | |
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\(\displaystyle{\int{\frac{-1}{x^2}dx}}\) | |
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solution |
Practice A05 | |
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\(\displaystyle{\int{\frac{3}{x^3}+2x^{3/2}-1~dx}}\) | |
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solution |
Practice A06 | |
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\(\displaystyle{\int{x^{5/2}-\frac{5}{x^4}-\sqrt{x}~dx}}\) | |
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solution |
Practice A07 | |
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\(\displaystyle{\int{\frac{3}{2}x^{1/2}+7~dx}}\) | |
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solution |
Practice A08 | |
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\(\displaystyle{\int{1+2x-4x^3~dx}}\) | |
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solution |
Practice A09 | |
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\(\displaystyle{\int{(x+1)(x^2+3)~dx}}\) | |
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solution |
Practice A10 | |
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\(\displaystyle{\int{-5e^x+7\sin(x)~dx}}\) | |
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solution |
Practice A11 | |
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\(\displaystyle{\int{2e^x-1+\sin(x)\csc(x)~dx}}\) | |
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solution |
Level B - Intermediate |
Practice B01 | |
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\(\displaystyle{\int{\frac{2}{x^{3/4}}-\frac{3}{x^{2/3}}~dx}}\) | |
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solution |
Practice B02 | |
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\(\displaystyle{\int{2x\sqrt{x}-\frac{1}{\sqrt{x}}dx}}\) | |
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solution |
Practice B03 | |
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\(\displaystyle{\int{\frac{3x-2}{\sqrt{x}}dx}}\) | |
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solution |
