Page 201 - Calculus Workbook For Dummies

185
Chapter 10: Integration: Reverse Differentiation

The Substitution Method:

Pulling the Switcheroo

The group of guess-and-check problems in the last section involve integrands that differ
from the standard integrand of a memorized antiderivative rule by a numerical amount.
The next set of problems involves integrands where the extra thing they contain
includes a variable expression. For these problems, you can still use the guess-and-check
method, but the traditional way of doing such problems is with the substitution method.

Q. Antidifferentiate # x 2 sinx dx with the sub- Q. Evaluate # 3 π 2 3
3
stitution method. x sinx dx.
0
A. - 1 cos x + C A. 2
3
3
3
1. This is the same as the previous Step 1
1. If a function in the integrand has some-
thing other than a plain old x for its except that at the same time as setting u
3
argument, set u equal to that argument. equal to x , you take the two x-indices
of integration and turn them into
u x 3 u-indices of integration.
=
2. Take the derivative of u with respect to Like this:
x, then throw the dx to the right side. 3
u x
=
du = 3 x 2
=
dx when x = 0 , u 0
3
2
=
du 3 x dx when x = 3 π , u = 3 π = π
3. Tweak your integrand so it contains the So 0 and π are the two u-indices of
result from Step 2 ( x dx3 2 ); and com- integration.
pensate for this tweak amount by multi-
plying the integral by the reciprocal of 2.–3. Steps 2-3 are identical to 2-3 in the pre-
the tweak number. vious example except that you happen
to be dealing with a definite integral in
x # 2 sinx dx this problem.
3
4. Pull the switcheroo. This time in addi-
You need a 3 in the integrand, so put in a 3 2
1 tion to replacing the x and the x dx3
3 and compensate with a . with their u-equivalents, you also
3
1 3 # 2 3 replace the x-indices with the u-indices:
= x sinx dx
3 = # π
1
1
= # sin x 3 x dx 3 sinu du
3
2
3 S \ 0
u du
5. Evaluate.
4. Pull the switcheroo. 1 π
= - cosuE
1
= # sinu du 3 0
3 1 2
1 =
= - ^ - 1 - h
5. Antidifferentiate by using the deriva- 3 3
tive of cos x- in reverse. If you prefer, you can skip determining
1 the u-indices of integration; just replace
= - cosu + C 3
3 the u with x like you did above in Step 6,
and then evaluate the definite integral
6. Get rid of the u by switching back to
the original expression. with the original indices of integration.
(Your calc teacher may not like this, how-
1
3
= - cosx + C ever, because it’s not the book method.)
3 3 π
1 3
= - cosx E
3
0
1 3 3
3
= - c cos π - cos0 m
3
1 2
1 =
= - ^ - 1 - h
3 3