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Other Strategies for Integration
Use a table of integrals 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 \(x=\text{tan}\ \theta ,\) we can arrive at
\[\int \frac{dx}{\sqrt{1+{x}^{2}}}=\text{ln}(x+\sqrt{{x}^{2}+1})+C.\]However, using \(x=\text{sinh}\ \theta ,\) we obtained a different solution, namely,
\[\int \frac{dx}{\sqrt{1+{x}^{2}}}={\text{sinh}}^{-1}x+C.\]We later showed algebraically that the two solutions are equivalent. That is, we showed that \({\text{sinh}}^{-1}x=\text{ln}(x+\sqrt{{x}^{2}+1}).\) 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
Try it.
Use the table formula
\[\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\frac{\sqrt{{a}^{2}-{u}^{2}}}{u}-{\text{sin}}^{-1}\frac{u}{a}+C\]to evaluate \(\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}dx.\)
Solution
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 \(du={e}^{x}\text{dx}.\) 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
\[\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}dx=\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{2x}}{e}^{x}dx.\]Substituting \(u={e}^{x}\) and \(du={e}^{x}\text{dx}\) produces \(\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du.\) From the integration table (#88 in Appendix A),
\[\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\frac{\sqrt{{a}^{2}-{u}^{2}}}{u}-{\text{sin}}^{-1}\frac{u}{a}+C.\]Thus,
\[\begin{array}{lllll}\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}dx & =\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{2x}}{e}^{x}dx & & & \text{Substitute}\ u={e}^{x}\ \text{and}\ du={e}^{x}dx. \\ & =\int \frac{\sqrt{{4}^{2}-{u}^{2}}}{{u}^{2}}du & & & \text{Apply the formula using}\ a=4. \\ & =-\frac{\sqrt{{4}^{2}-{u}^{2}}}{u}-{\text{sin}}^{-1}\frac{u}{4}+C & & & \text{Substitute}\ u={e}^{x}. \\ & =-\frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}-{\text{sin}}^{-1}(\frac{{e}^{x}}{4})+C. & & & \end{array}\]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
Try it.
Use a computer algebra system to evaluate \(\int \frac{dx}{\sqrt{{x}^{2}-4}}.\) 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.
Solution
Using Wolfram Alpha, we obtain
\[\int \frac{dx}{\sqrt{{x}^{2}-4}}=\text{ln}|\sqrt{{x}^{2}-4}+x|+C.\]Notice that
\[\text{ln}|\frac{\sqrt{{x}^{2}-4}}{2}+\frac{x}{2}|+C=\text{ln}|\frac{\sqrt{{x}^{2}-4}+x}{2}|+C=\text{ln}|\sqrt{{x}^{2}-4}+x|-\text{ln}\ 2+C.\]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.
Example
Try it.
Evaluate \({\int }^{\text{}}{\text{sin}}^{3}x\ dx\) using a CAS. Compare the result to \(\frac{1}{3}{\text{cos}}^{3}x-\text{cos}\ x+C,\) the result we might have obtained using the technique for integrating odd powers of \(\text{sin}\ x\) discussed earlier in this chapter.
Solution
Using Wolfram Alpha, we obtain
\[\int {\text{sin}}^{3}x\ dx=\frac{1}{12}(\text{cos}(3x)-9\ \text{cos}\ x)+C.\]This looks quite different from \(\frac{1}{3}{\text{cos}}^{3}x-\text{cos}\ x+C.\) To see that these antiderivatives are equivalent, we can make use of a few trigonometric identities:
\[\begin{array}{ll}\frac{1}{12}(\text{cos}(3x)-9\ \text{cos}\ x) & =\frac{1}{12}(\text{cos}(x+2x)-9\ \text{cos}\ x) \\ & =\frac{1}{12}(\text{cos}(x)\text{cos}(2x)-\text{sin}(x)\text{sin}(2x)-9\ \text{cos}\ x) \\ & =\frac{1}{12}(\text{cos}\ x(2\ {\text{cos}}^{2}x-1)-\text{sin}\ x(2\ \text{sin}\ x\ \text{cos}\ x)-9\ \text{cos}\ x) \\ & =\frac{1}{12}(2{\text{cos}}^{3}x-\text{cos}\ x-2\ \text{cos}\ x(1-{\text{cos}}^{2}x)-9\ \text{cos}\ x) \\ & =\frac{1}{12}(4{\text{cos}}^{3}x-12\ \text{cos}\ x) \\ & =\frac{1}{3}{\text{cos}}^{3}x-\text{cos}\ x.\end{array}\]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.
Key Concepts
- An integration table may be used to evaluate indefinite integrals.
- A CAS (or computer algebra system) may be used to evaluate indefinite integrals.
- It may require some effort to reconcile equivalent solutions obtained using different methods.
Other Strategies for Integration
Use a table of integrals to evaluate the following integrals.
Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.
Use a calculator or CAS to evaluate the following integrals.
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.
Use tables to perform the integration.
ഇപ്പോള് നീ ഒരു കോംപൌണ്ടും ഇത് ഉറപ്പിക്കുന്നില്ല, പക്ഷേ അതിന്റെ കഷ്ണങ്ങള് ചേര്ന്നു തീർക്കാന് പറ്റും. താഴെയൊന്ന് ശ്രമിക്കൂ അല്ലെങ്കില് നിങ്ങള്ക്കുതന്നെ ടൈപ്പ് ചെയ്യുക.
അഭ്യാസം (40)
ആദ്യം ഓരോന്നും കടലാസില് പരീക്ഷിച്ചുനോക്കുക. ഉത്തരം വെളിപ്പെടുത്തുക; ഓരോ പടിയിലും അവ തുറക്കാം.
-
Use the table formula
\[\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\frac{\sqrt{{a}^{2}-{u}^{2}}}{u}-{\text{sin}}^{-1}\frac{u}{a}+C\]to evaluate \(\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}dx.\)
ഉത്തരം വെളിപ്പെടുത്തുക
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 \(du={e}^{x}\text{dx}.\) 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
\[\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}dx=\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{2x}}{e}^{x}dx.\]Substituting \(u={e}^{x}\) and \(du={e}^{x}\text{dx}\) produces \(\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du.\) From the integration table (#88 in Appendix A),
\[\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\frac{\sqrt{{a}^{2}-{u}^{2}}}{u}-{\text{sin}}^{-1}\frac{u}{a}+C.\]Thus,
\[\begin{array}{lllll}\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}dx & =\int \frac{\sqrt{16-{e}^{2x}}}{{e}^{2x}}{e}^{x}dx & & & \text{Substitute}\ u={e}^{x}\ \text{and}\ du={e}^{x}dx. \\ & =\int \frac{\sqrt{{4}^{2}-{u}^{2}}}{{u}^{2}}du & & & \text{Apply the formula using}\ a=4. \\ & =-\frac{\sqrt{{4}^{2}-{u}^{2}}}{u}-{\text{sin}}^{-1}\frac{u}{4}+C & & & \text{Substitute}\ u={e}^{x}. \\ & =-\frac{\sqrt{16-{e}^{2x}}}{{e}^{x}}-{\text{sin}}^{-1}(\frac{{e}^{x}}{4})+C. & & & \end{array}\] -
Use a computer algebra system to evaluate \(\int \frac{dx}{\sqrt{{x}^{2}-4}}.\) 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.
ഉത്തരം വെളിപ്പെടുത്തുക
Using Wolfram Alpha, we obtain
\[\int \frac{dx}{\sqrt{{x}^{2}-4}}=\text{ln}|\sqrt{{x}^{2}-4}+x|+C.\]Notice that
\[\text{ln}|\frac{\sqrt{{x}^{2}-4}}{2}+\frac{x}{2}|+C=\text{ln}|\frac{\sqrt{{x}^{2}-4}+x}{2}|+C=\text{ln}|\sqrt{{x}^{2}-4}+x|-\text{ln}\ 2+C.\]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 \({\int }^{\text{}}{\text{sin}}^{3}x\ dx\) using a CAS. Compare the result to \(\frac{1}{3}{\text{cos}}^{3}x-\text{cos}\ x+C,\) the result we might have obtained using the technique for integrating odd powers of \(\text{sin}\ x\) discussed earlier in this chapter.
ഉത്തരം വെളിപ്പെടുത്തുക
Using Wolfram Alpha, we obtain
\[\int {\text{sin}}^{3}x\ dx=\frac{1}{12}(\text{cos}(3x)-9\ \text{cos}\ x)+C.\]This looks quite different from \(\frac{1}{3}{\text{cos}}^{3}x-\text{cos}\ x+C.\) To see that these antiderivatives are equivalent, we can make use of a few trigonometric identities:
\[\begin{array}{ll}\frac{1}{12}(\text{cos}(3x)-9\ \text{cos}\ x) & =\frac{1}{12}(\text{cos}(x+2x)-9\ \text{cos}\ x) \\ & =\frac{1}{12}(\text{cos}(x)\text{cos}(2x)-\text{sin}(x)\text{sin}(2x)-9\ \text{cos}\ x) \\ & =\frac{1}{12}(\text{cos}\ x(2\ {\text{cos}}^{2}x-1)-\text{sin}\ x(2\ \text{sin}\ x\ \text{cos}\ x)-9\ \text{cos}\ x) \\ & =\frac{1}{12}(2{\text{cos}}^{3}x-\text{cos}\ x-2\ \text{cos}\ x(1-{\text{cos}}^{2}x)-9\ \text{cos}\ x) \\ & =\frac{1}{12}(4{\text{cos}}^{3}x-12\ \text{cos}\ x) \\ & =\frac{1}{3}{\text{cos}}^{3}x-\text{cos}\ x.\end{array}\]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.
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Use a CAS to evaluate \(\int \frac{dx}{\sqrt{{x}^{2}+4}}.\)
ഉത്തരം വെളിപ്പെടുത്തുക
Possible solutions include \({\text{sinh}}^{-1}(\frac{x}{2})+C\) and \(\text{ln}|\sqrt{{x}^{2}+4}+x|+C.\)
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\(\int _{0}^{4}\frac{x}{\sqrt{1+2x}}dx\)
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\(\int \frac{x+3}{{x}^{2}+2x+2}dx\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(\frac{1}{2}\text{ln}|{x}^{2}+2x+2|+2\ \text{arctan}(x+1)+C\)
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\(\int {x}^{3}\sqrt{1+2{x}^{2}}\ dx\)
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\(\int \frac{1}{\sqrt{{x}^{2}+6x}}dx\)
ഉത്തരം വെളിപ്പെടുത്തുക
\({\text{cosh}}^{-1}(\frac{x+3}{3})+C\)
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\(\int \frac{x}{x+1}dx\)
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\(\int x\cdot {2}^{{x}^{2}}dx\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(\frac{{2}^{{x}^{2}-1}}{\text{ln}\ 2}+C\)
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\(\int \frac{1}{4{x}^{2}+25}dx\)
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\(\int \frac{dy}{\sqrt{4-{y}^{2}}}\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(\text{arcsin}(\frac{y}{2})+C\)
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\(\int {\text{sin}}^{3}(2x)\text{cos}(2x)dx\)
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\(\int \text{csc}(2w)\text{cot}(2w)dw\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(-\frac{1}{2}\text{csc}(2w)+C\)
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\(\int {2}^{y}dy\)
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\({\int }_{0}^{1}\frac{3x\ dx}{\sqrt{{x}^{2}+8}}\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(9-6\sqrt{2}\)
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\({\int }_{-1\text{/}4}^{1\text{/}4}{\text{sec}}^{2}(\pi x)\text{tan}(\pi x)dx\)
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\({\int }_{0}^{\pi \text{/}2}{\text{tan}}^{2}(\frac{x}{2})dx\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(2-\frac{\pi }{2}\)
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\(\int {\text{cos}}^{3}x\ dx\)
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\(\int {\text{tan}}^{5}(3x)dx\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(\frac{1}{12}{\text{tan}}^{4}(3x)-\frac{1}{6}{\text{tan}}^{2}(3x)+\frac{1}{3}\text{ln}|\text{sec}(3x)|+C\)
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\(\int {\text{sin}}^{2}y\ {\text{cos}}^{3}ydy\)
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[T] \(\int \frac{dw}{1+\text{sec}(\frac{w}{2})}\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(2\ \text{cot}(\frac{w}{2})-2\ \text{csc}(\frac{w}{2})+w+C\)
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[T] \(\int \frac{dw}{1-\text{cos}(7w)}\)
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[T] \({\int }_{0}^{t}\frac{dt}{4\ \text{cos}\ t+3\ \text{sin}\ t}\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(\frac{1}{5}\text{ln}|\frac{2(5+4\ \text{sin}\ t-3\ \text{cos}\ t)}{4\ \text{cos}\ t+3\ \text{sin}\ t}|\)
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[T] \(\int \frac{\sqrt{{x}^{2}-9}}{3x}dx\)
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[T] \(\int \frac{dx}{{x}^{1\text{/}2}+{x}^{1\text{/}3}}\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(6{x}^{1\text{/}6}-3{x}^{1\text{/}3}+2\sqrt{x}-6\ \text{ln}[1+{x}^{1\text{/}6}]+C\)
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[T] \(\int \frac{dx}{x\sqrt{x-1}}\)
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[T] \(\int {x}^{3}\text{sin}\ x\ dx\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(\text{-}{x}^{3}\text{cos}\ x+3{x}^{2}\text{sin}\ x+6x\ \text{cos}\ x-6\ \text{sin}\ x+C\)
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[T] \(\int x\sqrt{{x}^{4}-9}\ dx\)
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[T] \(\int \frac{x}{1+{e}^{\text{-}{x}^{2}}}dx\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(\frac{1}{2}({x}^{2}+\text{ln}|1+{e}^{\text{-}{x}^{2}}|)+C\)
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[T] \(\int \frac{\sqrt{3-5x}}{2x}dx\)
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[T] \(\int \frac{dx}{x\sqrt{x-1}}\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(2\ \text{arctan}(\sqrt{x-1})+C\)
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[T] \(\int {e}^{x}{\text{cos}}^{-1}({e}^{x})dx\)
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[T] \({\int }_{0}^{\pi \text{/}4}\text{cos}(2x)dx\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(0.5=\frac{1}{2}\)
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[T] \({\int }_{0}^{1}x\cdot {e}^{\text{-}{x}^{2}}dx\)
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[T] \({\int }_{0}^{8}\frac{2x}{\sqrt{{x}^{2}+36}}dx\)
ഉത്തരം വെളിപ്പെടുത്തുക
8.0
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[T] \({\int }_{0}^{2\text{/}\sqrt{3}}\frac{1}{4+9{x}^{2}}dx\)
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[T] \(\int \frac{dx}{{x}^{2}+4x+13}\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(\frac{1}{3}\text{arctan}(\frac{1}{3}(x+2))+C\)
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[T] \(\int \frac{dx}{1+\text{sin}\ x}\)
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\(\int \frac{dx}{{x}^{2}+2x+10}\)
ഉത്തരം വെളിപ്പെടുത്തുക
\(\frac{1}{3}\text{arctan}(\frac{x+1}{3})+C\)
ഒരു സ്വതന്ത്ര അക്കൗണ്ട്, നിങ്ങൾ പഠിച്ച കാര്യങ്ങൾ, നിങ്ങളുടെ പ്രശ്നങ്ങൾ പരിഹരിച്ചിരിക്കുന്ന പ്രശ്നങ്ങൾ, ഒരു സ്ഥലത്ത് ഒരു ടീച്ചർ, ഈ പേജിനെ കുറിച്ച് ചോദിക്കാവുന്നതാണ്.
മുകളില് ഒപ്പ് വയ്ക്കുക ലോഗിന്ഇവിടെ ഉപയോഗിച്ചിരിക്കുന്ന ചിഹ്നങ്ങള്
ഒരു പ്രത്യേക പത്രികയിൽ, ഒരു ചിത്രത്തിന്റെ പ്രതീകം, അതിലുള്ള എല്ലാ അക്ഷരങ്ങൾ എന്നിവയ്ക്കും വേണ്ടിയുള്ളതാണ്.
എങ്ങനെ: Other Strategies for Integration
- Use a table of integrals to solve integration problems.
- Use a computer algebra system (CAS) to solve integration problems.
- An integration table may be used to evaluate indefinite integrals.
- A CAS (or computer algebra system) may be used to evaluate indefinite integrals.
- It may require some effort to reconcile equivalent solutions obtained using different methods.
ആളുകൾ ചോദിക്കുന്നു
What is a derivative in one sentence?
The slope of the graph at a point, the rate at which the output is changing there. Speed is the derivative of position.
What is an integral in one sentence?
The accumulated total of a rate: the area under the curve. Distance travelled is the integral of speed.
Why are derivatives and integrals opposites?
That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.
When do I use substitution and when integration by parts?
Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function, a polynomial times an exponential, log or trig function.
ഈ താളിലെ ഭാഗങ്ങൾ മാറ്റുന്നു OpenStax Calculus Volume 2 (CC BY-NC-SA 4.0). ഇവിടെ വളച്ചൊടിച്ച്, വീണ്ടും പ്രഹരം, തെറ്റുകള് നമ്മുടേതാണ്. കാരണം അതിന്റെ ഉറവിടം ലൈസന്സ് ആണ്. കാരണം, ഈ പാഠം സ്വതന്ത്രമാണ്, വിലയില്ലാത്തതാണ്, എല്ലാം ഒരേ ലൈസന്സ് ആണ്.
കൂടുതല് Calculus
LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests