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

However, using [latex]x=\text{sinh}\theta[/latex], we obtained a different solution—namely,

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.

Watch the following video to see the worked solution to Example: Using a Formula from a Table to Evaluate an Integral

You can view the transcript for “3.5.1” here (opens in new window).

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: Using a CAS to Evaluate an Integral

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.

Show Solution

Using Wolfram Alpha, we obtain

[latex]\displaystyle\int {\sin}^{3}xdx=\frac{1}{12}\left(\cos\left(3x\right)-9\cos{x}\right)+C[/latex].

 

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:

[latex]\begin{array}{cc}\hfill \frac{1}{12}\left(\cos\left(3x\right)-9\cos{x}\right)& =\frac{1}{12}\left(\cos\left(x+2x\right)-9\cos{x}\right)\hfill \\ & =\frac{1}{12}\left(\cos\left(x\right)\cos\left(2x\right)-\sin\left(x\right)\sin\left(2x\right)-9\cos{x}\right)\hfill \\ & =\frac{1}{12}\left(\cos{x}\left(2{\cos}^{2}x - 1\right)-\sin{x}\left(2\sin{x}\cos{x}\right)-9\cos{x}\right)\hfill \\ & =\frac{1}{12}\left(2{\cos}^{3}x-\cos{x} - 2\cos{x}\left(1-{\cos}^{2}x\right)-9\cos{x}\right)\hfill \\ & =\frac{1}{12}\left(4{\cos}^{3}x - 12\cos{x}\right)\hfill \\ & =\frac{1}{3}{\cos}^{3}x-\cos{x}.\hfill \end{array}[/latex]

 

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.

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

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “3.5.2” here (opens in new window).