Page 207 - Calculus Workbook For Dummies

191
Chapter 10: Integration: Reverse Differentiation

u # 5 x e dx = 5 e + C
4
4
x
3
x
4
4
=
1. It’s not e plain old x , so u x .
2. You know the drill: du 4 x dx
3
=
3. Tweak: # 4 x e dx
5
4
x
3
4
4. Switch: = # e du
5
u
4
5
u
5. Antidifferentiate: = e + C
4
5 4
x
6. Switch back: = e + C
4
v # sec 2 x dx 2 tan x + C
=
x
2
1. It’s not sec plain old xi, so u = x .
_
1 - / 1 2 1
2. Differentiate: du = x dx = dx
2 2 x
3. Tweak: 2 # sec 2 x dx
2 x
4. Switch: 2 # sec u du
2
=
5. Antidifferentiate: 2 tanu + C
=
6. Switch back: 2 tan x + C
=
2
w # tdt 4 . .0011
2
0 _ t + 5i
1. Do the U and Diff (it’s sweeping the nation!), and find the u-indices of integration.
2
u t + 5 when t 0 , u 5
=
=
=
du 2 tdt when t 2 , u 9
=
=
=
2
1
2. Two tweaks: = # 2 tdt
2 2 4
0 _ t + 5i
9
3. The switch: = # du
1
2 u 4 9
5
1 1 1
-
3
-
3
3
-
4. Antidifferentiate and evaluate: = $ c - m u G = - _ 9 - 5 i . .0011
2 3 6
3 5
/ 2 3
8 _ s + 5i
x # ds 1974 .375
=
3 s
1
2 2
/ 2 3
You know the drill: u s + 5 ; du = s - / 1 3 ds = ds
=
3 3 3 s
3 3
8 _ s + 5i 8 2 _ s + 5i
/ 2 3
/ 2 3
# ds = # ds
3
3 s 2 3 3 s
1 1
8
3 # u du
3
2
1