Page 230 - Calculus Workbook For Dummies

214 Part IV: Integration and Infinite Series

i
2
i
6. Integrate: = # _ sec i - 1i d = tan - i + C .
7. Switch back to x (look at Step 4).
2
2
1
= 4 x - - arctan 4 x - + C
1
or
2
= 4 x - - arcsec x2 + C
1
s # 5 dx = 5 ln 2 x - 1 + C
2
2 x + 7 x - 4 9 x + 4
1. Factor the denominator: = # 5 dx .
2 ^ x - 1 ^h x + 4h
5 A B
2. Break up the fraction into a sum of partial fractions: = + .
2 ^ x - 1 ^h x + 4h 2 x - 1 x + 4
x
3. Multiply both sides by the least common denominator: 5 = A x + h B 2 - 1h.
4 +
^
^
4. Plug the roots of the factors into x one at a time.
1
4
x = - gives you x = gives you
2
9
5 = - 9 B 5 = A
2
5
B = - A = 10
9 9
5. Split up your integral and integrate.
# 5 dx = 10 # dx + - 5 # dx = 5 ln x - 1 - 5 ln x + 4 + C = 5 ln 2 x - 1 + C
2
2
2 x + 7 x - 4 9 2 x - 1 9 x + 4 9 9 9 x + 4
t # 2 x - 3 = - 7 ln x - 1 + 11 ln x + 4 - 13 ln x + 5 + C .
3
3 ^ x - 1 ^h x + 4 ^h x + 5h 208 13 16
1. The denominator is already factored, so go ahead and write your sum of partial fractions.
2 x - 3 = A + B + C
3 ^ x - 1 ^h x + 4 ^h x + 5h 3 x - 1 x + 4 x + 5
2. Multiply both sides by the LCD.
2 x - = A x + 4 ^h x + h B 3 - 1 ^h x + h C 3 - 1 ^h x + 4h
x
x
3
5 +
5 +
^
^
^
3. Plug the roots of the factors into x one at a time.
1 7 208 21
x = gives you : - = ; A A = -
3 3 9 208
11
x = - 4 " " " : - 11 = - 13 ; B B =
13
13
x = - 5 " " " : - 13 = 16 C C = -
16
4. Split up and integrate.
# 2 x - 3 dx = - 21 # dx + 11 # dx + - 13 # dx
3 ^ x - 1 ^h x + 4 ^h x + 5h 208 3 x - 1 13 x + 4 16 x + 5
- 7 11 13
3
= ln x - 1 + ln x + 4 - ln x + 5 + C
208 13 16