Evaluate the given integral
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Control-drag the integral and press the Enter key.
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Context Panel: Simplify≻Simplify
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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 secants and tangents.)
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Use the value command to evaluate the integral, or follow the approach in Table 6.3.17(b), below.
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Revert the change of variables by applying the substitution .
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From Figure 6.3.3, , , and . Note how the simplification isolates the additive constant of integration, .
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 6.3.17(a).
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Table 6.3.17(a) The substitution made by the Integration Methods tutor
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Note that the solution in Table 6.3.17(a) is not complete - the antiderivatives have not been obtained and the Revert rule has not been applied. Moreover, the Rewrite rule has to be used where, at top level, the expand command would have sufficed.
Table 6.3.17(b) shows the result when the Change rule is imposed on the tutor.
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Table 6.3.17(b) Integration Methods tutor after is imposed
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To put the integrand into the form of a multiple of , the Rewrite rule would have to be applied. Maple's stepwise code instead applied the additional change of variable, , which actually does lead to a solution because the problem has now become the equivalent of Example 6.3.15.
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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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