Page 222 - Calculus Workbook For Dummies

206 Part IV: Integration and Infinite Series

3. Apply the “7” rule.
1
x # arctan x dx = 1 x 2 arctan x - # x dx
2
2 2 1 + x 2
2
1
1 1 x + - 1
= x 2 arctan x - # dx
2 2 1 + x 2
1
1
1 2 arctan x - # dx + # dx
= x
2 2 2 1 + x 2
2
1 2 1 1 x + 1 x
= x arctan x - x + arctan x + C or arctan x - + C
2 2 2 2 2
1
d # x10 x dx = 101 ln10 - 99
2
10^ ln10h
- 1
1. Pick the algebraic x as your u.
2. Box it.
x 10 x
ln10
diff int
x
dx 10 dx

3. Do the “7”.
1
1 x 1
# x10 x dx = x10 F - 1 # 10 x dx
ln10 ln10
- 1 - 1 - 1
1
10 1 1 10 x
= + - $ F
ln10 10 ln10 ln10 ln10
- 1
10 1 10 1
= + - 2 +
ln10 10 ln10 ^ ln10h 10^ ln10h 2
101 ln10 - 99
= 2
10^ ln10h
*e # x e dx =- e _ x + 2 x + i C
2
x
-
2
x
-
2 +
2
1. Pick x as your u.
2. Box it.
x 2 –e –x
diff int
–x
2xdx e dx

3. “7” it: # x e dx = - x e + # xe dx
-
x
2
x
x
-
2
-
2
In the second integral, the power of x is reduced by 1, so you’re making progress.