Page 223 - Calculus Workbook For Dummies

207
Chapter 11: Integration Rules for Calculus Connoisseurs

4. Repeat the process for the second integral: Pick it and box it.

x –e –x

diff int
–x
dx e dx

5. “7” rule for the second integral: # xe dx = - xe + # e dx = - xe - e + C
-
-
x
-
x
x
-
x
x
-
6. Take this result and plug it into the second integral from Step 3.
# x e dx = - x e + _ xe - e + C i
-
x
-
x
-
2
-
2
x
x
2 -
x
2
-
-
x
x
-
2
-
x
2 +
= - x e - 2 xe - 2 e + C = - e _ x + 2 x + i C
*f # e x sin x dx = e x sin x - e x cos x + C
2 2
1. Pick sin x as your u — it’s a T from LIATE.
2. Box it.
sinx e x
diff int
x
cosxdx e dx
3. “7” it: # e x sin x dx e x sin x - # e x cosx dx
=
Doesn’t look like progress, but it is. Repeat this process for # e x cosx dx.
4. Pick cos x as your u and box it.
cosx e x
diff int
x
–sinxdx e dx
5. “7” it: # e x cosx dx e x cosx + # e x sin x dx
=
The prodigal son returns home and is rewarded.
6. Plug this result into the second integral from Step 3.
e # x sin x dx e x sin x - e x cosx - # e x sin x dx
=