### Learning Outcomes

- 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

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.

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

Use the table formula

to evaluate [latex]\displaystyle\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}dx[/latex].

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

### Try It

## 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 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.

### Interactive

You can access this integral calculator for more practice calculating integrals.

### 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.

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).

### try it

Use a CAS to evaluate [latex]\displaystyle\int \frac{dx}{\sqrt{{x}^{2}+4}}[/latex].