1.5 Substitution

The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. The drawback of this method, though, is that we must be able to find an antiderivative, and this is not always easy. In this section we examine a technique, called integration by substitution    , to help us find antiderivatives. Specifically, this method helps us find antiderivatives when the integrand is the result of a chain-rule derivative.

At first, the approach to the substitution procedure may not appear very obvious. However, it is primarily a visual task—that is, the integrand shows you what to do; it is a matter of recognizing the form of the function. So, what are we supposed to see? We are looking for an integrand of the form $f\left[g\left(x\right)\right]g^{\prime }\text{(}x)dx.$ For example, in the integral ${\displaystyle \int {\left(x^2-3\right)}^{3}2xdx},$ we have $f(x)=x^3,g(x)=x^2-3,$ and $g\text{'}(x)=2x.$ Then,

and we see that our integrand is in the correct form.

The method is called substitution because we substitute part of the integrand with the variable u and part of the integrand with du . It is also referred to as change of variables    because we are changing variables to obtain an expression that is easier to work with for applying the integration rules.

Proof

Let f , g , u , and F be as specified in the theorem. Then

Integrating both sides with respect to x , we see that

If we now substitute $u=g\left(x\right),$ and $du=g\text{'}\left(x\right)dx,$ we get

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Returning to the problem we looked at originally, we let $u=x^2-3$ and then $du=2xdx.$ Rewrite the integral in terms of u :

Using the power rule for integrals, we have

Substitute the original expression for x back into the solution:

We can generalize the procedure in the following Problem-Solving Strategy.