Chapter 6: Techniques of Integration
Section 6.6: Rationalizing Substitutions
Evaluate the indefinite integral ∫xx−1 ⅆx.
The rationalizing substitution u=x−1 (from Table 6.6.1) changes ∫xx−1 ⅆx to ∫u+1u ⅆu. Then
= ∫u1/2+u−1/2 ⅆu
Control-drag the given integral.
Context Panel: Evaluate and Display Inline
∫xx−1 ⅆx = 23⁢x−1⁢x+2
Table 6.6.1(a) contains the solution generated by the
tutor when the Constant, Constant Multiple, and Sum rules are taken as Understood Rules. Note the substitution made by the tutor: it is consistent with Table 6.6.1, but it is not the one made in the Mathematical Solution, above.
Table 6.6.1(a) Stepwise solution via Integration Methods tutor
Table 6.6.1(b) provides a solution based on the Change command in the IntegrationTools package. This solution illustrates a work-flow that might be used in a calculation that arises outside the context of the classroom.
Install the IntegrationTools package.
Assign the given integral the name q.
Context Panel: 2-D Math≻Convert To≻Inert Form
Define and name the substitution.
Apply the Change command from the IntegrationTools package.
Apply the value command to evaluate the inert form of the new integral.
Expression palette: Evaluation template
(Revert the substitution.)
Ax=a|f(x)TX = 23⁢x−13/2+2⁢x−1
Table 6.6.1(b) Rationalizing substitution via the Change command in the IntegrationTools package
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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