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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.
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 $x=\text{tan}\phantom{\rule{0.1em}{0ex}}\theta ,$ we can arrive at
However, using $x=\text{sinh}\phantom{\rule{0.1em}{0ex}}\theta ,$ we obtained a different solution—namely,
We later showed algebraically that the two solutions are equivalent. That is, we showed that ${\text{sinh}}^{\mathrm{-1}}x=\text{ln}\left(x+\sqrt{{x}^{2}+1}\right).$ 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.
Use the table formula
to evaluate $\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}dx.$
If we look at integration tables, we see that several formulas contain expressions of the form $\sqrt{{a}^{2}-{u}^{2}}.$ This expression is actually similar to $\sqrt{16-{e}^{2x}},$ where $a=4$ and $u={e}^{x}.$ Keep in mind that we must also have $du={e}^{x}.$ Multiplying the numerator and the denominator of the given integral by ${e}^{x}$ should help to put this integral in a useful form. Thus, we now have
Substituting $u={e}^{x}$ and $du={e}^{x}$ produces $\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du.$ From the integration table (#88 in Appendix A ),
Thus,
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.
Use a computer algebra system to evaluate $\int \frac{dx}{\sqrt{{x}^{2}-4}}.$ Compare this result with $\text{ln}\left|\frac{\sqrt{{x}^{2}-4}}{2}+\frac{x}{2}\right|+C,$ a result we might have obtained if we had used trigonometric substitution.
Using Wolfram Alpha, we obtain
Notice that
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.
Evaluate ${{\displaystyle \int}}^{\text{}}{\text{sin}}^{3}x\phantom{\rule{0.1em}{0ex}}dx$ using a CAS. Compare the result to $\frac{1}{3}{\text{cos}}^{3}x-\text{cos}\phantom{\rule{0.1em}{0ex}}x+C,$ the result we might have obtained using the technique for integrating odd powers of $\text{sin}\phantom{\rule{0.1em}{0ex}}x$ discussed earlier in this chapter.
Using Wolfram Alpha, we obtain
This looks quite different from $\frac{1}{3}{\text{cos}}^{3}x-\text{cos}\phantom{\rule{0.1em}{0ex}}x+C.$ To see that these antiderivatives are equivalent, we can make use of a few trigonometric identities:
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.
Use a CAS to evaluate $\int \frac{dx}{\sqrt{{x}^{2}+4}}.$
Possible solutions include ${\text{sinh}}^{\mathrm{-1}}\left(\frac{x}{2}\right)+C$ and $\text{ln}\left|\sqrt{{x}^{2}+4}+x\right|+C.$
Use a table of integrals to evaluate the following integrals.
$\underset{0}{\overset{4}{\int}}\frac{x}{\sqrt{1+2x}}dx$
$\int \frac{x+3}{{x}^{2}+2x+2}dx$
$\frac{1}{2}\text{ln}\left|{x}^{2}+2x+2\right|+2\phantom{\rule{0.1em}{0ex}}\text{arctan}\left(x+1\right)+C$
$\int {x}^{3}\sqrt{1+2{x}^{2}}\phantom{\rule{0.1em}{0ex}}dx$
$\int \frac{1}{\sqrt{{x}^{2}+6x}}dx$
${\text{cosh}}^{\mathrm{-1}}\left(\frac{x+3}{3}\right)+C$
$\int \frac{x}{x+1}dx$
$\int x\xb7{2}^{{x}^{2}}dx$
$\frac{{2}^{{x}^{2}-1}}{\text{ln}\phantom{\rule{0.1em}{0ex}}2}+C$
$\int \frac{1}{4{x}^{2}+25}dx$
$\int \frac{dy}{\sqrt{4-{y}^{2}}}$
$\text{arcsin}\left(\frac{y}{2}\right)+C$
$\int {\text{sin}}^{3}(2x)\text{cos}(2x)dx$
$\int \text{csc}(2w)\text{cot}(2w)dw$
$-\frac{1}{2}\text{csc}\left(2w\right)+C$
$\int {2}^{y}dy$
$\int}_{0}^{1}\frac{3x\phantom{\rule{0.1em}{0ex}}dx}{\sqrt{{x}^{2}+8}$
$9-6\sqrt{2}$
${\int}_{\mathrm{-1}\text{/}4}^{1\text{/}4}{\text{sec}}^{2}(\pi x)\text{tan}\left(\pi x\right)dx$
${\int}_{0}^{\pi \text{/}2}{\text{tan}}^{2}\left(\frac{x}{2}\right)dx$
$2-\frac{\pi}{2}$
$\int {\text{cos}}^{3}x\phantom{\rule{0.1em}{0ex}}dx$
$\int {\text{tan}}^{5}\left(3x\right)dx$
$\frac{1}{12}{\text{tan}}^{4}\left(3x\right)-\frac{1}{6}{\text{tan}}^{2}\left(3x\right)+\frac{1}{3}\text{ln}\left|\text{sec}(3x)\right|+C$
$\int {\text{sin}}^{2}y\phantom{\rule{0.1em}{0ex}}{\text{cos}}^{3}ydy$
Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.
[T] $\int \frac{dw}{1+\text{sec}\left(\frac{w}{2}\right)}$
$2\phantom{\rule{0.1em}{0ex}}\text{cot}\left(\frac{w}{2}\right)-2\phantom{\rule{0.1em}{0ex}}\text{csc}\left(\frac{w}{2}\right)+w+C$
[T] $\int \frac{dw}{1-\text{cos}\left(7w\right)}$
[T] $\int}_{0}^{t}\frac{dt}{4\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t+3\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t$
$\frac{1}{5}\text{ln}\left|\frac{2(5+4\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t-3\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t)}{4\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t+3\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t}\right|$
[T] $\int \frac{\sqrt{{x}^{2}-9}}{3x}dx$
[T] $\int \frac{dx}{{x}^{1\text{/}2}+{x}^{1\text{/}3}}$
$6{x}^{1\text{/}6}-3{x}^{1\text{/}3}+2\sqrt{x}-6\phantom{\rule{0.1em}{0ex}}\text{ln}\left[1+{x}^{1\text{/}6}\right]+C$
[T] $\int \frac{dx}{x\sqrt{x-1}}$
[T] $\int {x}^{3}\text{sin}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}dx$
$\text{\u2212}{x}^{3}\text{cos}\phantom{\rule{0.1em}{0ex}}x+3{x}^{2}\text{sin}\phantom{\rule{0.1em}{0ex}}x+6x\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x-6\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x+C$
[T] $\int x\sqrt{{x}^{4}-9}\phantom{\rule{0.1em}{0ex}}dx$
[T] $\int \frac{x}{1+{e}^{\text{\u2212}{x}^{2}}}dx$
$\frac{1}{2}\left({x}^{2}+\text{ln}\left|1+{e}^{\text{\u2212}{x}^{2}}\right|\right)+C$
[T] $\int \frac{\sqrt{3-5x}}{2x}dx$
[T] $\int \frac{dx}{x\sqrt{x-1}}$
$2\phantom{\rule{0.1em}{0ex}}\text{arctan}\left(\sqrt{x-1}\right)+C$
[T] $\int {e}^{x}{\text{cos}}^{\mathrm{-1}}({e}^{x})dx$
Use a calculator or CAS to evaluate the following integrals.
[T] ${\int}_{0}^{\pi \text{/}4}\text{cos}(2x)dx$
$0.5=\frac{1}{2}$
[T] ${\int}_{0}^{1}x\xb7{e}^{\text{\u2212}{x}^{2}}dx$
[T] ${\int}_{0}^{2\text{/}\sqrt{3}}\frac{1}{4+9{x}^{2}}dx$
[T] $\int \frac{dx}{{x}^{2}+4x+13}$
$\frac{1}{3}\text{arctan}\left(\frac{1}{3}(x+2)\right)+C$
[T] $\int \frac{dx}{1+\text{sin}\phantom{\rule{0.1em}{0ex}}x}$
Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table.
$\int \frac{dx}{{x}^{2}+2x+10}$
$\frac{1}{3}\text{arctan}\left(\frac{x+1}{3}\right)+C$
$\int \frac{dx}{\sqrt{{x}^{2}-6x}}$
$\int \frac{{e}^{x}}{\sqrt{{e}^{2x}-4}}dx$
$\text{ln}\left({e}^{x}+\sqrt{4+{e}^{2x}}\right)+C$
$\int \frac{\text{cos}\phantom{\rule{0.1em}{0ex}}x}{{\text{sin}}^{2}x+2\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x}dx$
$\int \frac{\text{arctan}\left({x}^{3}\right)}{{x}^{4}}dx$
$\text{ln}\phantom{\rule{0.1em}{0ex}}x-\frac{1}{6}\text{ln}\left({x}^{6}+1\right)-\frac{\text{arctan}\left({x}^{3}\right)}{3{x}^{3}}+C$
$\int \frac{\text{ln}\left|x\right|\text{arcsin}\left(\text{ln}\left|x\right|\right)}{x}dx$
Use tables to perform the integration.
$\int \frac{dx}{\sqrt{{x}^{2}+16}}$
$\text{ln}\left|x+\sqrt{16+{x}^{2}}\right|+C$
$\int \frac{3x}{2x+7}dx$
$\int \frac{dx}{1-\text{cos}\left(4x\right)}$
$-\frac{1}{4}\text{cot}(2x)+C$
$\int \frac{dx}{\sqrt{4x+1}}$
Find the area bounded by $y\left(4+25{x}^{2}\right)=5,x=0,y=0,\text{and}\phantom{\rule{0.2em}{0ex}}x=4.$ Use a table of integrals or a CAS.
$\frac{1}{2}\text{arctan}\phantom{\rule{0.1em}{0ex}}10$
The region bounded between the curve $y=\frac{1}{\sqrt{1+\text{cos}\phantom{\rule{0.1em}{0ex}}x}},0.3\le x\le 1.1,$ and the x -axis is revolved about the x -axis to generate a solid. Use a table of integrals to find the volume of the solid generated. (Round the answer to two decimal places.)
Use substitution and a table of integrals to find the area of the surface generated by revolving the curve $y={e}^{x},0\le x\le 3,$ about the x -axis. (Round the answer to two decimal places.)
1276.14
[T] Use an integral table and a calculator to find the area of the surface generated by revolving the curve $y=\frac{{x}^{2}}{2},0\le x\le 1,$ about the x -axis. (Round the answer to two decimal places.)
[T] Use a CAS or tables to find the area of the surface generated by revolving the curve $y=\text{cos}\phantom{\rule{0.1em}{0ex}}x,0\le x\le \frac{\pi}{2},$ about the x -axis. (Round the answer to two decimal places.)
7.21
Find the length of the curve $y=\frac{{x}^{2}}{4}$ over $\left[0,8\right].$
Find the length of the curve $y={e}^{x}$ over $\left[0,\text{ln}(2)\right].$
$\sqrt{5}-\sqrt{2}+\text{ln}\left|\frac{2+2\sqrt{2}}{1+\sqrt{5}}\right|$
Find the area of the surface formed by revolving the graph of $y=2\sqrt{x}$ over the interval $\left[0,9\right]$ about the x -axis.
Find the average value of the function $f(x)=\frac{1}{{x}^{2}+1}$ over the interval $\left[\mathrm{-3},3\right].$
$\frac{1}{3}\text{arctan}\left(3\right)\approx 0.416$
Approximate the arc length of the curve $y=\text{tan}\left(\pi x\right)$ over the interval $\left[0,\frac{1}{4}\right].$ (Round the answer to three decimal places.)
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