Chapter 6: Techniques of Integration
Section 6.5: Integrating the Fractions in a Partial-Fraction Decomposition
Example 6.5.2
Evaluate the integral ∫7x3−86x2+346x−455x4−15x3+84x2−208 x+192 ⅆx.
Solution
Mathematical Solution
From the partial-fraction decomposition in Example 6.4.3, it follows that
∫7x3−86x2+346x−455x4−15x3+84x2−208 x+192 ⅆx
= ∫2x−3ⅆx+∫5x−4ⅆx−∫7x−42ⅆx+∫1x−43ⅆx
=2 lnx−3+5 ln(|x−4|)+7x−4−1−x−4−2/2
Maple Solution
Evaluation in Maple
Control-drag the given integral.
Context Panel: Evaluate and Display Inline
∫7x3−86x2+346x−455x4−15x3+84x2−208 x+192 ⅆx = 5lnx−4+7x−4−12x−42+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 6.5.2(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. For each of the partial fractions, the tutor insists on an making an explicit change of variables to implement what would be an obvious integration.
∫7x3−86x2+346x−455x4−15x3+84x2−208x+192ⅆx=−7∫1x−42ⅆx+∫1x−43ⅆx+5∫1x−4ⅆx+2∫1x−3ⅆxpartialfractions
Table 6.5.2(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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