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Mathematics LibreTexts

7.5: Other Strategies for Integration

  • Gilbert Strang & Edwin “Jed” Herman
  • OpenStax

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Learning Objectives
  • Use a table of integrals to solve integration problems.
  • Use a computer algebra system (CAS) to solve integration problems.

In addition to the techniques of integration we have already seen, several other tools are widely available to assist with the process of integration. Among these tools are integration tables, which are readily available in many books, including the appendices to this one. Also widely available are computer algebra systems (CAS), which are found on calculators and in many campus computer labs, and are free online.

Tables of Integrals

Integration tables, if used in the right manner, can be a handy way either to evaluate or check an integral quickly. Keep in mind that when using a table to check an answer, it is possible for two completely correct solutions to look very different. For example, in Trigonometric Substitution, we found that, by using the substitution x=tanθ, we can arrive at

dx1+x2=ln|x+x2+1|+C.

However, using x=sinhθ, we obtained a different solution—namely,

dx1+x2=sinh1x+C.

We later showed algebraically that the two solutions are equivalent. That is, we showed that sinh1x=ln|x+x2+1|. In this case, the two antiderivatives that we found were actually equal. This need not be the case. However, as long as the difference in the two antiderivatives is a constant, they are equivalent.

Example 7.5.1: Using a Formula from a Table to Evaluate an Integral

Use the table formula

a2u2u2du=a2u2usin1ua+C

to evaluate 16e2xexdx.

Solution

If we look at integration tables, we see that several formulas contain expressions of the form a2u2. This expression is actually similar to 16e2x, where a=4 and u=ex. Keep in mind that we must also have du=ex. Multiplying the numerator and the denominator of the given integral by ex should help to put this integral in a useful form. Thus, we now have

16e2xexdx=16e2xe2xexdx.

Substituting u=ex and du=exdx produces a2u2u2du. From the integration table (#88 in Appendix A),

a2u2u2du=a2u2usin1ua+C.

Thus,

16e2xexdx=16e2xe2xexdx Substitute u=ex and du=exdx.

=42u2u2du Apply the formula using a=4.

=42u2usin1u4+C Substitute u=ex.

=16e2xexsin1(ex4)+C

Computer Algebra Systems

If available, a CAS is a faster alternative to a table for solving an integration problem. Many such systems are widely available and are, in general, quite easy to use.

Example 7.5.2: Using a Computer Algebra System to Evaluate an Integral

Use a computer algebra system to evaluate dxx24. Compare this result with ln|x242+x2|+C, a result we might have obtained if we had used trigonometric substitution.

Solution

Using Wolfram Alpha, we obtain

dxx24=ln|x24+x|+C.

Notice that

ln|x242+x2|+C=ln|x24+x2|+C=ln|x24+x|ln2+C.

Since these two antiderivatives differ by only a constant, the solutions are equivalent. We could have also demonstrated that each of these antiderivatives is correct by differentiating them.

You can access an integral calculator for more examples.

Example 7.5.3: Using a CAS to Evaluate an Integral

Evaluate sin3xdx using a CAS. Compare the result to 13cos3xcosx+C, the result we might have obtained using the technique for integrating odd powers of sinx discussed earlier in this chapter.

Solution

Using Wolfram Alpha, we obtain

sin3xdx=112(cos(3x)9cosx)+C.

This looks quite different from 13cos3xcosx+C. To see that these antiderivatives are equivalent, we can make use of a few trigonometric identities:

112(cos(3x)9cosx)=112(cos(x+2x)9cosx)

=112(cos(x)cos(2x)sin(x)sin(2x)9cosx)

=112(cosx(2cos2x1)sinx(2sinxcosx)9cosx)

=112(2cos3xcosx2cosx(1cos2x)9cosx)

=112(4cos3x12cosx)

=13cos3xcosx.

Thus, the two antiderivatives are identical.

We may also use a CAS to compare the graphs of the two functions, as shown in the following figure.

This is the graph of a periodic function. The waves have an amplitude of approximately 0.7 and a period of approximately 10. The graph represents the functions y = cos^3(x)/3 – cos(x) and y = 1/12(cos(3x)-9cos(x). The graph is the same for both functions.
Figure 7.5.1: The graphs of y=13cos3xcosx and y=112(cos(3x)9cosx) are identical.
Exercise 7.5.1

Use a CAS to evaluate dxx2+4.

Hint

Answers may vary.

Answer

Possible solutions include sinh1(x2)+C and ln|x2+4+x|+C.

Key Concepts

  • An integration table may be used to evaluate indefinite integrals.
  • A CAS (or computer algebra system) may be used to evaluate indefinite integrals.
  • It may require some effort to reconcile equivalent solutions obtained using different methods.

Glossary

computer algebra system (CAS)
technology used to perform many mathematical tasks, including integration
integration table
a table that lists integration formulas

This page titled 7.5: Other Strategies for Integration is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

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