Evaluate the given integral
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Control-drag the integral.
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Context Panel: Evaluate and Display Inline
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=
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Using the appropriate identity in Table 2.10.4, the alternate form of the solution, namely,
can be obtained from the Maple solution.
A stepwise solution that uses top-level commands except for one application of the Change command from the IntegrationTools package:
Initialization
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Install the IntegrationTools package.
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Let be the name of the given integral.
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Change variables as per Table 6.3.1
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Use the Change command to apply the change of variables .
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Simplify the radical to . Note the restriction imposed on .
(Maple believes that the sine and cosine functions are "simpler" than tangents and secants.)
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Use the value command to evaluate the integral, or follow the approach in Table 6.3.15(b), below.
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To revert the change of variables, apply the substitution via
Context Panel: Evaluate at a Point≻
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Control-drag the first two terms and press the Enter key.
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Select these two terms (in the output) and select "normal" in the Smart Pup-Up at the top of the Context Panel.
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Control-drag the result and to it, control-drag the log-term from the previous cell.
Press the Enter key.
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From Figure 6.3.2, , and .
The stepwise solution provided by the
tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules begins with the substitution and proceeds as shown in Table 6.3.15(a).
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Table 6.3.15(a) The substitution made by the Integration Methods tutor
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Note how the tutor requires the Rewrite rule where at top-level the expand command would suffice. Note further that the solution in Table 6.3.18 is not complete - the Revert rule has not been applied. If it were, the result would be the following, a result in dire need of a simplification that cannot be effected in the tutor.
Table 6.3.15(b) shows the result when the Change rule is imposed on the tutor.
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Table 6.3.15(b) Initial steps in an annotated stepwise solution via Integration Methods tutor
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Note the use of the Rewrite rule to effect transformations that at top-level could be implemented more directly. Note also that the two integrals in the last line of Table 6.3.15(b) are not evaluated. Maple's stepwise evaluation of these two integrals reproduces the derivations in Example 6.2.5 and Table 6.2.10, respectively.
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Note that an annotated stepwise solution is available via the Context Panel with the "All Solution Steps" option.
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The rules of integration can also be applied via the Context Panel, as per the figure to the right.
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