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17calculus > integrals

Integrals and Integration

The idea of integration is that integration 'undoes' the derivative, i.e. integration is the 'inverse' of differentiation. Now, I used quotes around the word 'inverse' because in actuality differentiation and integration are mostly inverses but not completely. Let's look at some notation first, before we get into the details.

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

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.

Okay, time for some practice problems.
When you are finished with those, we suggest integration by substitution as your next topic.

next: integration by substitution →

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faq: Why do integrals always have a dx?

Krista King Math - Why do integrals always have a dx? [4min-36secs]

Practice Problems

Instructions - - Unless otherwise instructed, evaluate the following integrals, giving your answers in exact form.

Level A - Basic

Practice A01

\(\displaystyle{\int{3x^2+2x+1~dx}}\)

solution

Practice A02

\(\displaystyle{\int{3x^4+5x-6~dx}}\)

solution

Practice A03

\(\displaystyle{\int{1-2x^2+3x^3~dx}}\)

solution

Practice A04

\(\displaystyle{\int{\frac{-1}{x^2}dx}}\)

solution

Practice A05

\(\displaystyle{\int{\frac{3}{x^3}+2x^{3/2}-1~dx}}\)

solution

Practice A06

\(\displaystyle{\int{x^{5/2}-\frac{5}{x^4}-\sqrt{x}~dx}}\)

solution

Practice A07

\(\displaystyle{\int{\frac{3}{2}x^{1/2}+7~dx}}\)

solution

Practice A08

\(\displaystyle{\int{1+2x-4x^3~dx}}\)

solution

Practice A09

\(\displaystyle{\int{(x+1)(x^2+3)~dx}}\)

solution

Practice A10

\(\displaystyle{\int{-5e^x+7\sin(x)~dx}}\)

solution

Practice A11

\(\displaystyle{\int{2e^x-1+\sin(x)\csc(x)~dx}}\)

solution


Level B - Intermediate

Practice B01

\(\displaystyle{\int{\frac{2}{x^{3/4}}-\frac{3}{x^{2/3}}~dx}}\)

solution

Practice B02

\(\displaystyle{\int{2x\sqrt{x}-\frac{1}{\sqrt{x}}dx}}\)

solution

Practice B03

\(\displaystyle{\int{\frac{3x-2}{\sqrt{x}}dx}}\)

solution

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