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Section5.3Integration by Substitution

Motivating Questions
  • How can we begin to find algebraic formulas for antiderivatives of more complicated algebraic functions?

  • What is an indefinite integral and how is its notation used in discussing antiderivatives?

  • How does the technique of \(u\)-substitution work to help us evaluate certain indefinite integrals, and how does this process rely on identifying function-derivative pairs?

In Section 4.4, we learned the key role that antiderivatives play in the process of evaluating definite integrals exactly. In particular, the Fundamental Theorem of Calculus tells us that if \(F\) is any antiderivative of \(f\text{,}\) then

\begin{equation*} \int_a^b f(x) \, dx = F(b) - F(a). \end{equation*}

Furthermore, we realized that each elementary derivative rule developed in Chapter 2 leads to a corresponding elementary antiderivative, as summarized in Table 4.4.4. Thus, if we wish to evaluate an integral such as

\begin{equation*} \int_0^1 \left(x^3 - \sqrt{x} + 5^x \right) \,dx, \end{equation*}

it is straightforward to do so, since we can easily antidifferentiate \(f(x) = x^3 - \sqrt{x} + 5^x.\) In particular, since a function \(F\) whose derivative is \(f\) is given by \(F(x) = \frac{1}{4}x^4 - \frac{2}{3}x^{3/2} + \frac{1}{\ln(5)}5^x\text{,}\) the Fundamental Theorem of Calculus tells us that

\begin{align*} \int_0^1 \left(x^3 - \sqrt{x} + 5^x\right) \,dx =\mathstrut \amp \left. \frac{1}{4}x^4 - \frac{2}{3}x^{3/2} + \frac{1}{\ln(5)}5^x\right|_0^1\\ =\mathstrut \amp \left( \frac{1}{4}(1)^4 - \frac{2}{3}(1)^{3/2} + \frac{1}{\ln(5)}5^1 \right) - \left( 0 - 0 + \frac{1}{\ln(5)}5^0 \right)\\ =\mathstrut \amp -\frac{5}{12} + \frac{4}{\ln(5)}. \end{align*}

Because an algebraic formula for an antiderivative of \(f\) enables us to evaluate the definite integral \(\int_a^b f(x) \, dx\) exactly, we see that we have a natural interest in being able to find such algebraic antiderivatives. Note that we emphasize algebraic antiderivatives, as opposed to any antiderivative, since we know by the Second Fundamental Theorem of Calculus that \(G(x) = \int_a^x f(t) \, dt\) is indeed an antiderivative of the given function \(f\text{,}\) but one that still involves a definite integral. One of our main goals in this section and the one following is to develop understanding, in select circumstances, of how to “undo” the process of differentiation in order to find an algebraic antiderivative for a given function.

Preview Activity5.3.1

In Section 2.5, we learned the Chain Rule and how it can be applied to find the derivative of a composite function. In particular, if \(u\) is a differentiable function of \(x\text{,}\) and \(f\) is a differentiable function of \(u(x)\text{,}\) then

\begin{equation*} \frac{d}{dx} \left[ f(u(x)) \right] = f'(u(x)) \cdot u'(x). \end{equation*}

In words, we say that the derivative of a composite function \(c(x) = f(u(x))\text{,}\) where \(f\) is considered the “outer” function and \(u\) the “inner” function, is “the derivative of the outer function, evaluated at the inner function, times the derivative of the inner function.”

  1. For each of the following functions, use the Chain Rule to find the function's derivative. Be sure to label each derivative by name (e.g., the derivative of \(g(x)\) should be labeled \(g'(x)\)).

    1. \(g(x) = e^{3x}\)

    2. \(h(x) = \sin(5x+1)\)

    3. \(p(x) = \arctan(2x)\)

    4. \(q(x) = (2-7x)^4\)

    5. \(r(x) = 3^{4-11x}\)

  2. For each of the following functions, use your work in (a) to help you determine the general antiderivative 1  of the function. Label each antiderivative by name (e.g., the antiderivative of \(m\) should be called \(M\)). In addition, check your work by computing the derivative of each proposed antiderivative.

    1. \(m(x) = e^{3x}\)

    2. \(n(x) = \cos(5x+1)\)

    3. \(s(x) = \frac{1}{1+4x^2}\)

    4. \(v(x) = (2-7x)^3\)

    5. \(w(x) = 3^{4-11x}\)

  3. Based on your experience in parts (a) and (b), conjecture an antiderivative for each of the following functions. Test your conjectures by computing the derivative of each proposed antiderivative.

    1. \(a(x) = \cos(\pi x)\)

    2. \(b(x) = (4x+7)^{11}\)

    3. \(c(x) = xe^{x^2}\)

Subsection5.3.1Reversing the Chain Rule: First Steps

In Preview Activity 5.3.1, we saw that it is usually straightforward to antidifferentiate a function of the form

\begin{equation*} h(x) = f(u(x)), \end{equation*}

whenever \(f\) is a familiar function whose antiderivative is known and \(u(x)\) is a linear function. For example, if we consider

\begin{equation*} h(x) = (5x-3)^6, \end{equation*}

in this context the outer function \(f\) is \(f(u) = u^6\text{,}\) while the inner function is \(u(x) = 5x - 3\text{.}\) Since the antiderivative of \(f\) is \(F(u) = \frac{1}{7}u^7+C\text{,}\) we see that the antiderivative of \(h\) is

\begin{equation*} H(x) = \frac{1}{7} (5x-3)^7 \cdot \frac{1}{5} + C = \frac{1}{35} (5x-3)^7 + C. \end{equation*}

The inclusion of the constant \(\frac{1}{5}\) is essential precisely because the derivative of the inner function is \(u'(x) = 5\text{.}\) Indeed, if we now compute \(H'(x)\text{,}\) we find by the Chain Rule (and Constant Multiple Rule) that

\begin{equation*} H'(x) = \frac{1}{35} \cdot 7(5x-3)^6 \cdot 5 = (5x-3)^6 = h(x), \end{equation*}

and thus \(H\) is indeed the general antiderivative of \(h\text{.}\)

Hence, in the special case where the outer function is familiar and the inner function is linear, we can antidifferentiate composite functions according to the following rule.

If \(h(x) = f(ax + b)\) and \(F\) is a known algebraic antiderivative of \(f\text{,}\) then the general antiderivative of \(h\) is given by

\begin{equation*} H(x) = \frac{1}{a} F(ax+b) + C. \end{equation*}

When discussing antiderivatives, it is often useful to have shorthand notation that indicates the instruction to find an antiderivative. Thus, in a similar way to how the notation

\begin{equation*} \frac{d}{dx} \left[ f(x) \right] \end{equation*}

represents the derivative of \(f(x)\) with respect to \(x\text{,}\) we use the notation of the indefinite integral,

\begin{equation*} \int f(x) \, dx \end{equation*}

to represent the general antiderivative of \(f\) with respect to \(x\text{.}\) For instance, returning to the earlier example with \(h(x) = (5x-3)^6\) above, we can rephrase the relationship between \(h\) and its antiderivative \(H\) through the notation

\begin{equation*} \int (5x-3)^6 \, dx = \frac{1}{35} (5x-6)^7 + C. \end{equation*}

When we find an antiderivative, we will often say that we evaluate an indefinite integral; said differently, the instruction to evaluate an indefinite integral means to find the general antiderivative. Just as the notation \(\frac{d}{dx} [ \Box ]\) means “find the derivative with respect to \(x\) of \(\Box\text{,}\)” the notation \(\int \Box \, dx\) means “find a function of \(x\) whose derivative is \(\Box\text{.}\)”

Activity5.3.2

Evaluate each of the following indefinite integrals. Check each antiderivative that you find by differentiating.

  1. \(\int \sin(8-3x) \, dx\)

  2. \(\int \sec^2 (4x) \, dx\)

  3. \(\int \frac{1}{11x - 9} \, dx\)

  4. \(\int \csc(2x+1) \cot(2x+1) \, dx\)

  5. \(\int \frac{1}{\sqrt{1-16x^2}}\, dx\)

  6. \(\int 5^{-x}\, dx\)

Subsection5.3.2Reversing the Chain Rule: \(u\)-substitution

Of course, a natural question arises from our recent work: what happens when the inner function is not a linear function? For example, can we find antiderivatives of such functions as

\begin{equation*} g(x) = x e^{x^2} \ \text{and} \ h(x) = e^{x^2}? \end{equation*}

It is important to explicitly remember that differentiation and antidifferentiation are essentially inverse processes; that they are not quite inverse processes is due to the \(+C\) that arises when antidifferentiating. This close relationship enables us to take any known derivative rule and translate it to a corresponding rule for an indefinite integral. For example, since

\begin{equation*} \frac{d}{dx} \left[x^5\right] = 5x^4, \end{equation*}

we can equivalently write

\begin{equation*} \int 5x^4 \, dx = x^5 + C. \end{equation*}

Recall that the Chain Rule states that

\begin{equation*} \frac{d}{dx} \left[ f(g(x)) \right] = f'(g(x)) \cdot g'(x). \end{equation*}

Restating this relationship in terms of an indefinite integral,

\begin{equation} \int f'(g(x)) g'(x) \, dx = f(g(x))+C. \tag{5.3.1} \end{equation}

Hence, Equation (5.3.1) tells us that if we can take a given function and view its algebraic structure as \(f'(g(x)) g'(x)\) for some appropriate choices of \(f\) and \(g\text{,}\) then we can antidifferentiate the function by reversing the Chain Rule. It is especially notable that both \(g(x)\) and \(g'(x)\) appear in the form of \(f'(g(x)) g'(x)\text{;}\) we will sometimes say that we seek to identify a function-derivative pair when trying to apply the rule in Equation (5.3.1).

In the situation where we can identify a function-derivative pair, we will introduce a new variable \(u\) to represent the function \(g(x)\text{.}\) Observing that with \(u = g(x)\text{,}\) it follows in Leibniz notation that \(\frac{du}{dx} = g'(x)\text{,}\) so that in terms of differentials 2 , \(du = g'(x)\, dx\text{.}\) Now converting the indefinite integral of interest to a new one in terms of \(u\text{,}\) we have

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

Provided that \(f'\) is an elementary function whose antiderivative is known, we can now easily evaluate the indefinite integral in \(u\text{,}\) and then go on to determine the desired overall antiderivative of \(f'(g(x)) g'(x)\text{.}\) We call this process \(u\)-substitution. To see \(u\)-substitution at work, we consider the following example.

Example5.3.1

Evaluate the indefinite integral

\begin{equation*} \int x^3 \cdot \sin (7x^4 + 3) \, dx \end{equation*}

and check the result by differentiating.

Solution

An essential observation about our work in Example 5.3.1 is that the \(u\)-substitution only worked because the function multiplying \(\sin (7x^4 + 3)\) was \(x^3\text{.}\) If instead that function was \(x^2\) or \(x^4\text{,}\) the substitution process may not (and likely would not) have worked. This is one of the primary challenges of antidifferentiation: slight changes in the integrand make tremendous differences. For instance, we can use \(u\)-substitution with \(u = x^2\) and \(du = 2xdx\) to find that

\begin{align*} \int xe^{x^2} \, dx =\mathstrut \amp \int e^u \cdot \frac{1}{2} \, du\\ =\mathstrut \amp \frac{1}{2} \int e^u \, du\\ =\mathstrut \amp \frac{1}{2} e^u + C\\ =\mathstrut \amp \frac{1}{2} e^{x^2} + C. \end{align*}

If, however, we consider the similar indefinite integral

\begin{equation*} \int e^{x^2} \, dx, \end{equation*}

the missing \(x\) to multiply \(e^{x^2}\) makes the \(u\)-substitution \(u = x^2\) no longer possible. Hence, part of the lesson of \(u\)-substitution is just how specialized the process is: it only applies to situations where, up to a missing constant, the integrand that is present is the result of applying the Chain Rule to a different, related function.

Activity5.3.3

Evaluate each of the following indefinite integrals by using these steps:

  • Find two functions within the integrand that form (up to a possible missing constant) a function-derivative pair;

  • Make a substitution and convert the integral to one involving \(u\) and \(du\text{;}\)

  • Evaluate the new integral in \(u\text{;}\)

  • Convert the resulting function of \(u\) back to a function of \(x\) by using your earlier substitution;

  • Check your work by differentiating the function of \(x\text{.}\) You should come up with the integrand originally given.

  1. \(\int \frac{x^2}{5x^3+1} \, dx\)

  2. \(\int e^x \sin(e^x) \, dx\)

  3. \(\int \frac{\cos(\sqrt{x})}{\sqrt{x}} \, dx\)

Subsection5.3.3Evaluating Definite Integrals via \(u\)-substitution

We have just introduced \(u\)-substitution as a means to evaluate indefinite integrals of functions that can be written, up to a constant multiple, in the form \(f(g(x))g'(x)\text{.}\) This same technique can be used to evaluate definite integrals involving such functions, though we need to be careful with the corresponding limits of integration. Consider, for instance, the definite integral

\begin{equation*} \int_2^5 xe^{x^2} \, dx. \end{equation*}

Whenever we write a definite integral, it is implicit that the limits of integration correspond to the variable of integration. To be more explicit, observe that

\begin{equation*} \int_2^5 xe^{x^2} \, dx = \int_{x=2}^{x=5} xe^{x^2} \, dx. \end{equation*}

When we execute a \(u\)-substitution, we change the variable of integration; it is essential to note that this also changes the limits of integration. For instance, with the substitution \(u = x^2\) and \(du = 2x \, dx\text{,}\) it also follows that when \(x = 2\text{,}\) \(u = 2^2 = 4\text{,}\) and when \(x = 5\text{,}\) \(u = 5^2 = 25.\) Thus, under the change of variables of \(u\)-substitution, we now have

\begin{align*} \int_{x=2}^{x=5} xe^{x^2} \, dx =\mathstrut \amp \int_{u=4}^{u=25} e^{u} \cdot \frac{1}{2} \, du\\ =\mathstrut \amp \left. \frac{1}{2}e^u \right|_{u=4}^{u=25}\\ =\mathstrut \amp \frac{1}{2}e^{25} - \frac{1}{2}e^4. \end{align*}

Alternatively, we could consider the related indefinite integral \(\int xe^{x^2} \, dx,\) find the antiderivative \(\frac{1}{2}e^{x^2}\) through \(u\)-substitution, and then evaluate the original definite integral. From that perspective, we'd have

\begin{align*} \int_{2}^{5} xe^{x^2} \, dx =\mathstrut \amp \left. \frac{1}{2}e^{x^2} \right|_{2}^{5}\\ =\mathstrut \amp \frac{1}{2}e^{25} - \frac{1}{2}e^4, \end{align*}

which is, of course, the same result.

Activity5.3.4

Evaluate each of the following definite integrals exactly through an appropriate \(u\)-substitution.

  1. \(\int_1^2 \frac{x}{1 + 4x^2} \, dx\)

  2. \(\int_0^1 e^{-x} (2e^{-x}+3)^{9} \, dx\)

  3. \(\int_{2/\pi}^{4/\pi} \frac{\cos\left(\frac{1}{x}\right)}{x^{2}} \,dx\)

Subsection5.3.4Summary

  • To begin to find algebraic formulas for antiderivatives of more complicated algebraic functions, we need to think carefully about how we can reverse known differentiation rules. To that end, it is essential that we understand and recall known derivatives of basic functions, as well as the standard derivative rules.

  • The indefinite integral provides notation for antiderivatives. When we write “\(\int f(x) \, dx\text{,}\)” we mean “the general antiderivative of \(f\text{.}\)” In particular, if we have functions \(f\) and \(F\) such that \(F' = f\text{,}\) the following two statements say the exact thing:

    \begin{equation*} \frac{d}{dx}[F(x)] = f(x) \ \text{and} \ \int f(x) \, dx = F(x) + C. \end{equation*}

    That is, \(f\) is the derivative of \(F\text{,}\) and \(F\) is an antiderivative of \(f\text{.}\)

  • The technique of \(u\)-substitution helps us to evaluate indefinite integrals of the form \(\int f(g(x))g'(x) \, dx\) through the substitutions \(u = g(x)\) and \(du = g'(x) \, dx\text{,}\) so that

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

    A key part of choosing the expression in \(x\) to be represented by \(u\) is the identification of a function-derivative pair. To do so, we often look for an “inner” function \(g(x)\) that is part of a composite function, while investigating whether \(g'(x)\) (or a constant multiple of \(g'(x)\)) is present as a multiplying factor of the integrand.

Subsection5.3.5Exercises

1
2
3
4
5
6
7

This problem centers on finding antiderivatives for the basic trigonometric functions other than \(\sin(x)\) and \(\cos(x)\text{.}\)

  1. Consider the indefinite integral \(\int \tan(x) \, dx\text{.}\) By rewriting the integrand as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) and identifying an appropriate function-derivative pair, make a \(u\)-substitution and hence evaluate \(\int \tan(x) \, dx\text{.}\)

  2. In a similar way, evaluate \(\int \cot(x) \, dx\text{.}\)

  3. Consider the indefinite integral

    \begin{equation*} \int \frac{\sec^2(x) + \sec(x) \tan(x)}{\sec(x) + \tan(x)} \, dx. \end{equation*}

    Evaluate this integral using the substitution \(u = \sec(x) + \tan(x)\text{.}\)

  4. Simplify the integrand in (c) by factoring the numerator. What is a far simpler way to write the integrand?

  5. Combine your work in (c) and (d) to determine \(\int \sec(x) \, dx\text{.}\)

  6. Using (c)-(e) as a guide, evaluate \(\int \csc(x) \, dx\text{.}\)

8

Consider the indefinite integral \(\int x \sqrt{x-1} \, dx.\)

  1. At first glance, this integrand may not seem suited to substitution due to the presence of \(x\) in separate locations in the integrand. Nonetheless, using the composite function \(\sqrt{x-1}\) as a guide, let \(u = x-1\text{.}\) Determine expressions for both \(x\) and \(dx\) in terms of \(u\text{.}\)

  2. Convert the given integral in \(x\) to a new integral in \(u\text{.}\)

  3. Evaluate the integral in (b) by noting that \(\sqrt{u} = u^{1/2}\) and observing that it is now possible to rewrite the integrand in \(u\) by expanding through multiplication.

  4. Evaluate each of the integrals \(\int x^2 \sqrt{x-1} \, dx\) and \(\int x \sqrt{x^2 - 1} \, dx\text{.}\) Write a paragraph to discuss the similarities among the three indefinite integrals in this problem and the role of substitution and algebraic rearrangement in each.

9

Consider the indefinite integral \(\int \sin^3(x) \, dx\text{.}\)

  1. Explain why the substitution \(u = \sin(x)\) will not work to help evaluate the given integral.

  2. Recall the Fundamental Trigonometric Identity, which states that \(\sin^2(x) + \cos^2(x) = 1\text{.}\) By observing that \(\sin^3(x) = \sin(x) \cdot \sin^2(x)\text{,}\) use the Fundamental Trigonometric Identity to rewrite the integrand as the product of \(\sin(x)\) with another function.

  3. Explain why the substitution \(u = \cos(x)\) now provides a possible way to evaluate the integral in (b).

  4. Use your work in (a)-(c) to evaluate the indefinite integral \(\int \sin^3(x) \, dx\text{.}\)

  5. Use a similar approach to evaluate \(\int \cos^3(x) \, dx\text{.}\)

10

For the town of Mathland, MI, residential power consumption has shown certain trends over recent years. Based on data reflecting average usage, engineers at the power company have modeled the town's rate of energy consumption by the function

\begin{equation*} r(t) = 4 + \sin(0.263t + 4.7) + \cos(0.526t+9.4). \end{equation*}

Here, \(t\) measures time in hours after midnight on a typical weekday, and \(r\) is the rate of consumption in megawatts 3  at time \(t\text{.}\) Units are critical throughout this problem.

  1. Sketch a carefully labeled graph of \(r(t)\) on the interval [0,24] and explain its meaning. Why is this a reasonable model of power consumption?

  2. Without calculating its value, explain the meaning of \(\int_0^{24} r(t) \, dt\text{.}\) Include appropriate units on your answer.

  3. Determine the exact amount of power Mathland consumes in a typical day.

  4. What is Mathland's average rate of energy consumption in a given 24-hour period? What are the units on this quantity?