### 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 $x=\tan\theta$, we can arrive at

$\displaystyle\int \frac{dx}{\sqrt{1+{x}^{2}}}=\text{ln}\left(x+\sqrt{{x}^{2}+1}\right)+C$.

However, using $x=\text{sinh}\theta$, we obtained a different solution—namely,

$\displaystyle\int \frac{dx}{\sqrt{1+{x}^{2}}}={\text{sinh}}^{-1}x+C$.

We later showed algebraically that the two solutions are equivalent. That is, we showed that ${\text{sinh}}^{-1}x=\text{ln}\left(x+\sqrt{{x}^{2}+1}\right)$. 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

$\displaystyle\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\frac{\sqrt{{a}^{2}-{u}^{2}}}{u}-{\sin}^{-1}\frac{u}{a}+C$

to evaluate $\displaystyle\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}dx$.

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 Computer Algebra System to Evaluate an Integral

Use a computer algebra system to evaluate $\displaystyle\int \frac{dx}{\sqrt{{x}^{2}-4}}$. Compare this result with $\text{ln}|\frac{\sqrt{{x}^{2}-4}}{2}+\frac{x}{2}|+C$, a result we might have obtained if we had used trigonometric substitution.

### Example: Using a CAS to Evaluate an Integral

Evaluate ${\displaystyle\int}{\sin}^{3}xdx$ using a CAS. Compare the result to $\frac{1}{3}{\cos}^{3}x-\cos{x}+C$, the result we might have obtained using the technique for integrating odd powers of $\sin{x}$ 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.

### try it

Use a CAS to evaluate $\displaystyle\int \frac{dx}{\sqrt{{x}^{2}+4}}$.