Substitution

Section 5.5 Substitution

Now the we understand a bit more about integration, we can start addressing the problem of evaluating integrals. Using the Fundamental Theorem of Calculus, we know that to evaluate definite integrals, we need to find an antiderivative of the integrand. But finding antiderivatives is not always obvious. In fact, it is a highly non-trivial problem in general; finding an antiderivative of a function is much more difficult than finding the derivative of a function, as the former is not algorithmic, while the latter is. To find the derivative of a function, you just need to apply repeatedly differentiation rules; but there is no precise recipe that I can give you to find antiderivatives of functions in general.

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However, all is not lost. There is a number of techniques that we can develop to help find antiderivatives of complicated functions. In this section we explore our first technique of integration, known as “substitution”. Other techniques of integration will be covered in MATH 146. Note that substitution is by far the most useful technique of integration: one that you will use over and over again in your mathematical life.

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Objectives

You should be able to:

Evaluate indefinite integrals using substitution.
Evaluate definite integrals using substitution and the Fundamental Theorem of Calculus.
Explain in what sense substitution undoes the chain rule.

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Subsection 5.5.1 Instructional video

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Subsection 5.5.2 Key concepts

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Concept 5.5.1. The substitution rule.

If $u=g(x)$ is a differentiable function and $f(x)$ is continuous over the range of $g(x)\text{,}$ then

\int f (g (x)) g^{'} (x) d x = \int f (u) d u .

$\begin{equation*} \int f(g(x)) g'(x)\ dx = \int f(u)\ du. \end{equation*}$

In other words, the substitution $u=g(x)\text{,}$ with $du = g'(x) dx\text{,}$ “undoes” the chain rule

In practice, what this means is that you can do a substitution $u=g(x)\text{,}$ $du = g'(x) dx$ inside an integral. This will be useful if you can then rewrite the integrand as a function of $u$ that is easier to integrate than the original integrand as a function of $x\text{.}$

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Concept 5.5.2. Substitution for definite integrals.

Substitution also works for definite integrals, but one has to be careful with the limits of integration. There are two methods to evaluate definite integrals using substitution. The first one is often the preferred method.

The idea is to transform the limits of integration from $x$ -values to $u$ -values as you perform the substitution:
$\begin{equation*} \int_a^b f((g(x)) g'(x)\ dx = \int_{g(a)}^{g(b)} f(u) \ du, \qquad \text{with }u=g(x), du = g'(x) dx. \end{equation*}$
Then you can evaluate the resulting definite integral in $u$ directly using the Fundamental Theorem of Calculus (i.e. by finding an antiderivative of $f(u)$ ).
The second method is to first find an antiderivative of the integrand using substitution, rewrite it in terms of the original variable $x\text{,}$ and then evaluate at the limits of integration $x=a$ and $x=b$ using the Fundamental Theorem of Calculus. This works as well, but you have to be careful with notation and make sure that you rewrite everything in terms of the $x$ -variable before you evaluate at the limits of integration.

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MATH 144: Calculus for the Physical Sciences I