Chapter 4: Integration
Section 4.4: Integration by Substitution
Evaluate the indefinite integral ∫2 x+18 ⅆx.
Set y=2 x+1 so that dy=2 dx and dx=dy/2. Under this change of variable, the given indefinite integral becomes
∫y8 dy2=12y99=2 x+19/18
Of course, there are settings in which the addition of an arbitrary constant is deemed essential.
Although it would be possible to expand the numerator and integrate termwise, it is not a recommended approach to integration! In fact, termwise integration produces a result that does not factor to the compact expression obtained above because of a missing additive constant.
∫2 x+18 ⅆx
= ∫256⁢x8+1024⁢x7+1792⁢x6+1792⁢x5+1120⁢x4+448⁢x3+112⁢x2+16⁢x+1 ⅆx
In fact, termwise integration produces a result that does not factor to the compact expression obtained above because of a missing additive constant of 1/18. This can be seen by expanding 2 x+19/18, that is, from
2 x+19/18= expand 2569⁢x9+128⁢x8+256⁢x7+8963⁢x6+224⁢x5+112⁢x4+1123⁢x3+8⁢x2+x+118
(which was done in Maple by applying Expand from the Context Panel.)
Annotated Stepwise Maple Solution
Figure 4.4.2(a) shows part of an annotated stepwise solution obtained with the
tutor. The steps are:
Change: Set u=2 x+1
Constant Multiple: Move the constant factor 1/2
Power Rule: Add 1 to the power and divide by the new power
Revert: Replace u with 2 x+1
Figure 4.4.2(a) Integration Methods tutor
The final step, Revert, isn't taken in Figure 4.4.2(a) because Maple expands 2 x+19/18, stretching the whole display and thereby putting the annotations "off the screen."
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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