Chapter 6: Techniques of Integration
Section 6.5: Integrating the Fractions in a Partial-Fraction Decomposition
Example 6.5.1
Evaluate the integral ∫7x−23x2−7 x+12 ⅆx.
Solution
Mathematical Solution
From the partial-fraction decomposition in Example 6.4.2, it follows that
∫7x−23x2−7 x+12 ⅆx
= ∫2x−3 ⅆx+∫5x−4 ⅆx
=2 lnx−3+5 ln(|x−4|)
Maple Solution
Evaluation in Maple
Control-drag the given integral.
Context Panel: Evaluate and Display Inline
∫7x−23x2−7 x+12 ⅆx = 5lnx−4+2lnx−3
Note once again that Maple integrates 1/x to lnx, not ln(x), relying on a complex constant of integration to counterbalance the logarithm of a negative number.
Table 6.5.1(a) shows the result of invoking the Partial Fractions rule in the tutor when the Sum and Constant Multiple rules are taken as Understood Rules. The next step the tutor insists on is the change of variables u=x−4, the outcome of which is the term 5 lnx−4; similarly for the integration of the other partial fraction.
∫7x−23x2−7x+12ⅆx=5∫1x−4ⅆx+2∫1x−3ⅆxpartialfractions
Table 6.5.1(a) Partial Fractions rule applied in Integration Methods tutor
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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