Chapter 6: Techniques of Integration
Section 6.6: Rationalizing Substitutions
Evaluate the indefinite integral ∫1x1/3+x1/4 ⅆx.
As per Table 6.6.1, make the substitution u=x1/12, or u12=x, so that dx=12 u11, and the given integral becomes
∫12 u11u4+u3 ⅆu
The expansion of the integrand in the second line is the result of a long division.
Maple's built-in integrator returns an unexpectedly long and cumbersome antiderivative. Obviously, the algorithm used to find this is not the one used by the
tutor, whose first step is shown in Table 6.6.2(a).
Table 6.6.2(a) First step taken in the Integration Methods tutor
The remaining steps of the integration are straightforward. However, Table 6.6.2(b) contains this solution implemented through the Change and GetIntegrand commands from the IntegrationTools package.
Install the IntegrationTools package.
Control-drag the given integral.
Context Panel: 2-D Math≻Convert To≻Inert Form
Apply the simplify command after changing variables with the Change command.
Effect a long division by imposing a partial-fraction decomposition on the integrand, which is extracted with the GetIntegrand command. Re-build the inert integral with the Int command.
Evaluate the integral with the value command.
Revert the original substitution.
q1≔simplifyChangeQ,x=u12,u assuming u>0
Table 6.6.2(b) Stepwise solution based on commands from the IntegrationTools package
Note that Maple's integrator applied to u8/1+u finds the same antiderivative whether or not the partial-fraction decomposition is imposed.
Note that an annotated stepwise solution is available via the Context Panel with the "All Solution Steps" option.
The rules of integration can also be applied via the Context Panel, as per the figure to the right.
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