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Section5.8Using Technology and Tables to Evaluate Integrals

Motivating Questions

  • What role have integral tables historically played in the study of calculus and how can a table be used to evaluate integrals such as \(\int \sqrt{a^2 + u^2} \, du\text{?}\)

  • What role can a computer algebra system play in the process of finding antiderivatives?

We have learned several antidifferentiation techniques: \(u\)-substitution, integration by parts, trigonometric substitution, and the method of partial fractions. However, there will still be cases of integrals that can not be solved by these methods. Consider the integral \(\int e^{x^2} \, dx\text{:}\) none of these methods can be used to solve this integral. In fact, an elementary algebraic antiderivative for \(e^{x^2}\) does not exist. No antidifferentiation method will provide us with a simple algebraic formula for a function \(F(x)\) that satisfies \(F'(x) = e^{x^2}\text{.}\)

The goal of this section is to recognize that there are many functions for which an algebraic formula for an antiderivative does not exist, and appreciate the role that computing technology can play in finding antiderivatives of other complicated functions. Moreover, tables of integrals can be helpful in finding antiderivatives quickly, even when we are able to compute them by hand.

Example5.74

For each of the indefinite integrals below, look at the table of integrals in AppendixA and decide how an antiderivative may be obtained using the table. It is also worthwhile to ask whether any of the methods we have learned so far can also be used to find an antiderivative.

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

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

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

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

Solution

  1. This can be evaluated by following Rule (a) in the table. It may also be evaluated using trigonometric substitution.

  2. This requires algebraic manipulation before you can successfully use a substitution. We can evaluate \(\int \frac{2x+3}{1+x^2} \, dx \) by recognizing that \(\frac{2x+3}{1+x^2} =\frac{2x}{1+x^2} + \frac{3}{1+x^2}\) so we can split the original integral into two integrals. Then, evaluate \(\int \frac{2x}{1+x^2} \, dx \) with the substitution \(u=1+x^2 \ , du=2x \, dx \) and evaluate \(\int \frac{3}{1+x^2} \, dx \) by following Rule (a) in the table or by using trigonometric substitution.

  3. This integral can be found by using the substitution \(u=e^x \ , du=e^x \, dx \) which transforms the integral to \(\int \frac{1}{1+u^2} \, du \text{.}\) The antiderivative of \(\frac{1}{1+u^2} \, du \) is again found using Rule (a) in the table; trigonometric substitution may also be used.

  4. The final integral can be evaluated by following Rule (g) in the table. It may also be evaluated using trigonometric substitution.

SubsectionUsing an Integral Table

Calculus has a rich history, spanning across many centuries and continents. Core concepts like limit and continuity can be traced back to Greek mathematicians, astronomers, and philosophers in 400-300 BC. Scholars in China, India, the Middle East, and Europe developed these ideas further over hundreds of years until the first unified accounts of differential and integral calculus were given, independently, by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century.

It is instructive to realize that because the personal computer didn't exist until the late 1980s, calculus (and all other mathematics) had to be done by hand for millenia. In the 21st century, however, computers have revolutionized many aspects of the world we live in, including mathematics. In this section we investigate the role that integral tables and computer algebra systems can play in evaluating indefinite integrals.

As seen in the short table of integrals found in AppendixA, there are many forms of integrals that involve \(\sqrt{a^2 \pm w^2}\) and \(\sqrt{w^2 - a^2}\text{.}\) These integral rules can be developed using the technique known as trigonometric substitution that we learned in the last section. To see how these rules are used, consider the differences among

\begin{equation*} \int \frac{1}{\sqrt{1-x^2}} \,dx, \ \ \ \int \frac{x}{\sqrt{1-x^2}} \,dx, \ \ \ \text{and} \ \ \ \int \sqrt{1-x^2} \,dx\text{.} \end{equation*}

The first integral is a familiar basic one, and results in \(\arcsin(x) + C\text{.}\) The second integral can be evaluated using a standard \(u\)-substitution with \(u = 1-x^2\text{.}\) The third, however, is not familiar and does not lend itself to \(u\)-substitution.

In AppendixA, we find the rule

\begin{equation*} \text{(h)} ~ \int \sqrt{a^2 - u^2} \, du = \frac{u}{2}\sqrt{a^2 - u^2} + \frac{a^2}{2} \arcsin \frac{u}{a} + C\text{.} \end{equation*}

Using the substitutions \(a = 1\) and \(u = x\) (so that \(du = dx\)), it follows that

\begin{equation*} \int \sqrt{1-x^2} \, dx = \frac{x}{2} \sqrt{1-x^2} + \frac{1}{2} \arcsin x + C\text{.} \end{equation*}

Whenever we are applying a rule in the table, we are doing a \(u\)-substitution, particularly when the substitution is more complicated than setting \(u = x\) as in the last example.

Example5.75

Evaluate the integral

\begin{equation*} \int \sqrt{9 + 64x^2} \, dx\text{.} \end{equation*}
Solution

Here, we want to use Rule (c) from the table, but we now set \(a = 3\) and \(u = 8x\text{.}\) We also choose the \(+\) option in the rule. With this substitution, it follows that \(du = 8dx\text{,}\) so \(dx = \frac{1}{8} du\text{.}\) Applying the substitution,

\begin{equation*} \int \sqrt{9 + 64x^2} \, dx = \int \sqrt{9 + u^2} \cdot \frac{1}{8} \, du = \frac{1}{8} \int \sqrt{9+u^2} \, du\text{.} \end{equation*}

By Rule (c), we now find that

\begin{align*} \int \sqrt{9 + 64x^2} \, dx =\mathstrut \amp \frac{1}{8} \left( \frac{u}{2}\sqrt{u^2 + 9} + \frac{9}{2}\ln\left|u + \sqrt{u^2 + 9}\right| + C \right)\\ =\mathstrut \amp \frac{1}{8} \left( \frac{8x}{2}\sqrt{64x^2 + 9} + \frac{9}{2}\ln\left|8x + \sqrt{64x^2 + 9}\right| + C \right)\text{.} \end{align*}

Whenever we use a \(u\)-subsitution in conjunction with AppendixA, it's important that we not forget to address any constants that arise and include them in our computations, such as the \(\frac{1}{8}\) that appeared in Example5.75.

Example5.76

For each of the following integrals, evaluate the integral using \(u\)-substitution and/or an entry from the table found in AppendixA.

  1. \(\int \sqrt{x^2 + 4} \, dx\)

  2. \(\int \frac{x}{\sqrt{x^2 +4}} \, dx\)

  3. \(\int \frac{2}{\sqrt{16+25x^2}}\, dx\)

  4. \(\int \frac{1}{x^2 \sqrt{49-36x^2}} \, dx\)

Hint

  1. Compare to \(\int \sqrt{u^2 + a^2} \, du\text{.}\)

  2. Try a straightforward \(u\)-substitution; the table is unneeded.

  3. Let \(a = 4\) and \(u = 5x\) and look for a similar integral in the table.

  4. Let \(a = 7\) and \(u = 6x\text{;}\) find a related integral in the table.

Answer

  1. \(\int \sqrt{x^2 + 4} \, dx = \frac{x}2\sqrt{x^2+4}+2\ln\big|x+\sqrt{x^2+4}\big|+C\text{.}\)

  2. \(\int \frac{x}{\sqrt{x^2 +4}} \, dx = \sqrt{x^2 + 4} + C\text{.}\)

  3. \(\int \frac{2}{\sqrt{16+25x^2}}\, dx = \frac{2}{5} \ln\big| 5x + \sqrt{16+25x^2} \big| + C\text{.}\)

  4. \(\int \frac{1}{x^2 \sqrt{49-36x^2}} \, dx = - \frac{\sqrt{49-36x^2}}{49x} + C\text{.}\)

Solution

  1. By Rule (c) in AppendixA with \(a=2\) and \(u = x\text{,}\)

    \begin{equation*} \int \sqrt{x^2 + 4} \, dx = \frac{x}2\sqrt{x^2+4}+\frac42\ln\left|x+\sqrt{x^2+4}\right|+C\text{.} \end{equation*}

  2. Let \(u=x^2 + 4\text{,}\) so \(du = 2x\, dx\text{.}\) Thus

    \begin{equation*} \int \frac{x}{\sqrt{x^2 +4}} \, dx = \frac{1}{2} \int \frac{du}{\sqrt{u}} = \frac{1}{2} \cdot 2u^{1/2} + C = \sqrt{x^2 + 4} + C\text{.} \end{equation*}

  3. Letting \(a = 4\) and \(u = 5x\text{,}\) we see \(du=5\,dx\text{.}\) Thus,

    \begin{equation*} \int \frac{2}{\sqrt{16+25x^2}}\, dx = \frac{2}{5} \int \frac{du}{\sqrt{a^2 + u^2}}\text{.} \end{equation*}

    Now by Rule (b) in AppendixA, it follows

    \begin{equation*} \int \frac{2}{\sqrt{16+25x^2}}\, dx = \frac{2}{5} \ln\left| 5x + \sqrt{16+25x^2} \right| + C\text{.} \end{equation*}

  4. Letting \(a = 7\) and \(u = 6x\text{,}\) it follows that \(x=\frac{1}{6}u\) and \(du=6\,dx\text{,}\) and therefore

    \begin{equation*} \int \frac{1}{x^2 \sqrt{49-36x^2}} \, dx = \frac{1}{6} \int \frac{1}{\frac{1}{36}u^2 \sqrt{a^2-u^2}} \, du\text{.} \end{equation*}

    Using Rule (k) in AppendixA and the fact that \(\frac{1}{6} \cdot 36 = 6\text{,}\) we see

    \begin{equation*} \int \frac{1}{x^2 \sqrt{49-36x^2}} \, dx = -6\cdot \frac{\sqrt{49-36x^2}}{49 \cdot 6x} + C\text{.} \end{equation*}

SubsectionUsing Computer Algebra Systems

A computer algebra system (CAS) is a computer program that is capable of executing symbolic mathematics. For example, if we ask a CAS to solve the equation \(ax^2 + bx + c = 0\) for the variable \(x\text{,}\) where \(a\text{,}\) \(b\text{,}\) and \(c\) are arbitrary constants, the program will return \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\text{.}\) Research to develop the first CAS dates to the 1960s, and these programs became publicly available in the early 1990s. Two prominent examples are the programs Maple and Mathematica, which were among the first computer algebra systems to offer a graphical user interface. Today, Maple and Mathematica are exceptionally powerful professional software packages that can execute an amazing array of sophisticated mathematical computations. They are also very expensive, as each is a proprietary program. The CAS SAGE is an open-source, free alternative to Maple and Mathematica.

For the purposes of this text, when we need to use a CAS, we are going to turn instead to a similar, but somewhat different computational tool, the web-based computational knowledge engine called WolframAlpha.12WolframAlpha can be accessed at www.wolframalpha.com. There are two features of WolframAlpha that make it stand out from the CAS options mentioned above: (1) unlike Maple and Mathematica, WolframAlpha is free (provided we are willing to navigate some pop-up advertising); and (2) unlike any of the three, the syntax in WolframAlpha is flexible. Think of WolframAlpha as being a little bit like doing a Google search: the program will interpret what is input, and then provide a summary of options.

If we want to have WolframAlpha evaluate an integral for us, we can provide it syntax such as

integrate x^2 dx

to which the program responds with

\begin{equation*} \int x^2 \, dx = \frac{x^3}{3} + \text{constant}\text{.} \end{equation*}

To find the partial fraction decomposition of any rational function, in WolframAlpha, entering

partial fraction 5x/(x^2-x-2)

results in the output

\begin{equation*} \frac{5x}{x^2-x-2} = \frac{10}{3(x-2)} + \frac{5}{3(x+1)}\text{.} \end{equation*}

While there is much to be enthusiastic about regarding CAS programs such as WolframAlpha, there are several things we should be cautious about: (1) a CAS only responds to exactly what is input; (2) a CAS can answer using powerful functions from very advanced mathematics; and (3) there are problems that even a CAS cannot do without additional human insight.

Although (1) likely goes without saying, we have to be careful with our input: if we enter syntax that defines the wrong function, the CAS will work with precisely the function we define. For example, if we are interested in evaluating the integral

\begin{equation*} \int \frac{1}{16-5x^2} \, dx\text{,} \end{equation*}

and we mistakenly enter

integrate 1/16 - 5x^2 dx

a CAS will (correctly) reply with

\begin{equation*} \frac{1}{16}x - \frac{5}{3} x^3\text{.} \end{equation*}

But if we are sufficiently well-versed in antidifferentiation, we will recognize that this function cannot be the one that we seek: integrating a rational function such as \(\frac{1}{16-5x^2}\text{,}\) we expect the logarithm function to be present in the result.

Regarding (2), even for a relatively simple integral such as \(\int \frac{1}{16-5x^2} \, dx\text{,}\) some CASs will invoke advanced functions rather than simple ones. For instance, if we use Maple to execute the command

int(1/(16-5*x^2), x);

the program responds with

\begin{equation*} \int \frac{1}{16-5x^2} \, dx = \frac{\sqrt{5}}{20} \arctanh \left(\frac{\sqrt{5}}{4}x\right)\text{.} \end{equation*}

While this is correct (save for the missing arbitrary constant, which Maple never reports), the inverse hyperbolic tangent function is not a common nor familiar one; a simpler way to express this function can be found by using the partial fractions method, and happens to be the result reported by WolframAlpha:

\begin{equation*} \int \frac{1}{16-5x^2} \, dx = \frac{1}{8\sqrt{5}} \left(\log(4\sqrt{5}+5x) - \log(4\sqrt{5}-5x)\right) + \text{constant}\text{.} \end{equation*}

Using sophisticated functions from more advanced mathematics is sometimes the way a CAS says to the user I don't know how to do this problem. For example, if we want to evaluate

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

and we ask WolframAlpha to do so, the input

integrate exp(-x^2) dx

results in the output

\begin{equation*} \int e^{-x^2} \, dx = \frac{\sqrt{\pi}}{2}\erf (x) + \text{constant}\text{.} \end{equation*}

The function erf\((x)\) is the error function, which is actually defined by an integral:

\begin{equation*} \erf (x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt\text{.} \end{equation*}

So, in producing an output involving an integral, the CAS has basically reported back to us the very question we asked.

Finally, as remarked at (3) above, there are times that a CAS will actually fail without some additional human insight. If we consider the integral

\begin{equation*} \int (1+x)e^x \sqrt{1+x^2e^{2x}} \, dx \end{equation*}

and ask WolframAlpha to evaluate

int (1+x) * exp(x) * sqrt(1+x^2 * exp(2x)) dx,

the program thinks for a moment and then reports

(no result found in terms of standard mathematical functions)

But in fact this integral is not that difficult to evaluate. If we let \(u = xe^{x}\text{,}\) then \(du = (1+x)e^x \, dx\text{,}\) which means that the preceding integral has the form

\begin{equation*} \int (1+x)e^x \sqrt{1+x^2e^{2x}} \, dx = \int \sqrt{1+u^2} \, du\text{,} \end{equation*}

which is a straightforward one for any CAS to evaluate.

So, we should proceed with some caution: while any CAS is capable of evaluating a wide range of integrals (both definite and indefinite), there are times when the result can mislead us. We must think carefully about the meaning of the output, whether it is consistent with what we expect, and whether or not it makes sense to proceed.

SubsectionSummary

  • Until the development of computer algebra systems, integral tables enabled students of calculus to evaluate integrals of certain forms quickly, such as \(\int \sqrt{a^2 + u^2} \, du\text{,}\) where \(a\) is a positive real number. A short table of integrals may be found in AppendixA.

  • Computer algebra systems can play an important role in finding antiderivatives, though we must be cautious to use correct input, to watch for unusual or unfamiliar advanced functions that the CAS may cite in its result, and to consider the possibility that a CAS may need further assistance or insight from us in order to answer a particular question.

SubsectionExercises

For each of the following integrals involving rational functions, (1) use a CAS to find the partial fraction decomposition of the integrand; (2) evaluate the integral of the resulting function without the assistance of technology; (3) use a CAS to evaluate the original integral to test and compare your result in (2).

  1. \(\int \frac{x^3 + x + 1}{x^4 - 1} \, dx\)

  2. \(\int \frac{x^5 + x^2 + 3}{x^3 - 6x^2 + 11x - 6} \, dx\)

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

For each of the following integrals involving radical functions, (1) use an appropriate \(u\)-substitution along with AppendixA to evaluate the integral without the assistance of technology, and (2) use a CAS to evaluate the original integral to test and compare your result in (1).

  1. \(\int \frac{1}{x \sqrt{9x^2 + 25}} \, dx\)

  2. \(\int x \sqrt{1 + x^4} \, dx\)

  3. \(\int e^x \sqrt{4 + e^{2x}} \, dx\)

  4. \(\int \frac{\tan(x)}{\sqrt{9 - \cos^2(x)}} \, dx\)

Consider the indefinite integral given by

\begin{equation*} \int \frac{\sqrt{x+\sqrt{1+x^2}}}{x} \, dx\text{.} \end{equation*}

  1. Explain why \(u\)-substitution does not offer a way to simplify this integral by discussing at least two different options you might try for \(u\text{.}\)

  2. Explain why integration by parts does not seem to be a reasonable way to proceed, either, by considering one option for \(u\) and \(dv\text{.}\)

  3. Is there any line in the integral table in AppendixA that is helpful for this integral?

  4. Evaluate the given integral using WolframAlpha. What do you observe?