Use a table of integrals to solve integration problems
Use a computer algebra system (CAS) to solve integration problems
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 [latex]x=\tan\theta [/latex], we can arrive at
We later showed algebraically that the two solutions are equivalent. That is, we showed that [latex]{\text{sinh}}^{-1}x=\text{ln}\left(x+\sqrt{{x}^{2}+1}\right)[/latex]. 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: Using a Formula from a Table to Evaluate an Integral
to evaluate [latex]\displaystyle\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}dx[/latex].
Show Solution
If we look at integration tables, we see that several formulas contain expressions of the form [latex]\sqrt{{a}^{2}-{u}^{2}}[/latex]. This expression is actually similar to [latex]\sqrt{16-{e}^{2x}}[/latex], where [latex]a=4[/latex] and [latex]u={e}^{x}[/latex]. Keep in mind that we must also have [latex]du={e}^{x}[/latex]. Multiplying the numerator and the denominator of the given integral by [latex]{e}^{x}[/latex] should help to put this integral in a useful form. Thus, we now have
Substituting [latex]u={e}^{x}[/latex] and [latex]du={e}^{x}[/latex] produces [latex]\displaystyle\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du[/latex]. From the integration table (#88 in Appendix A),
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: Using a Computer Algebra System to Evaluate an Integral
Use a computer algebra system to evaluate [latex]\displaystyle\int \frac{dx}{\sqrt{{x}^{2}-4}}[/latex]. Compare this result with [latex]\text{ln}|\frac{\sqrt{{x}^{2}-4}}{2}+\frac{x}{2}|+C[/latex], a result we might have obtained if we had used trigonometric substitution.
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.
Evaluate [latex]{\displaystyle\int}{\sin}^{3}xdx[/latex] using a CAS. Compare the result to [latex]\frac{1}{3}{\cos}^{3}x-\cos{x}+C[/latex], the result we might have obtained using the technique for integrating odd powers of [latex]\sin{x}[/latex] discussed earlier in this chapter.
This looks quite different from [latex]\frac{1}{3}{\cos}^{3}x-\cos{x}+C[/latex]. To see that these antiderivatives are equivalent, we can make use of a few trigonometric identities:
We may also use a CAS to compare the graphs of the two functions, as shown in the following figure.
Figure 1. The graphs of [latex]y=\frac{1}{3}{\cos}^{3}x-\cos{x}[/latex] and [latex]y=\frac{1}{12}\left(\cos\left(3x\right)-9\cos{x}\right)[/latex] are identical.
Watch the following video to see the worked solution to Example: Using a CAS to Evaluate an Integral
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