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 cosine function is "simpler" than the secant. )
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Use the value command to evaluate the integral, or follow the approach in Table 6.3.23(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 . This solution differs from the previous one by an 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 6.3.23(a).
The change of variables selected by the tutor leads to a multiple of , provided , and this corresponds to . Since the tutor does not have provision for routine simplifications, it takes several steps, including invocation of the Rewrite rule to massage the expression into a form where the power rule of integration applies.
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Table 6.3.23(a) The substitution made by the Integration Methods tutor
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Table 6.3.23(b) shows the result when the Change rule is imposed on the tutor.
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Table 6.3.23(b) Integration Methods tutor after is imposed
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It takes the Rewrite rule to remove the absolute value in the integrand, and without such, the tutor can go no further. The antiderivative of is derived in Example 6.2.5.
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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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