Page 231 - Calculus Workbook For Dummies

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Chapter 11: Integration Rules for Calculus Connoisseurs

2
u # x + + 1 dx = ln x - 1 - 3 2 ^ x - 1h 2 + C
x
2
3
x - 3 x + 3 x - 1 2^ x - 1h
2
1. Factor the denominator: = # x + + 1 dx.
x
^ x - 1h 3
2
x + + 1 A B C
x
2. Write the partial fractions: 3 = + 2 + 3 .
^ x - 1h x - 1 ^ x - 1h ^ x - 1h
2
2
1
x
1 +
3. Multiply by the LCD: x + + = A x - h B x - h C .
1 +
^
^
4. Plug in the single root, which is 1, giving you C = 3.
5. Equate coefficients of like terms.
2
Without multiplying out the entire right side in Step 3, you can see that the x term on the
2
2
right will be Ax . Because the coefficient of x on the left is 1, A must equal 1.
6. Plug in 0 for x, giving you 1 = A – B + C.
Because you know A is 1 and C is 3, B must be 3.
Note: You can solve for A, B, and C in many ways, but the way I did it is probably the quickest.
7. Split up and integrate.
2
x
# x + + 1 dx = # dx + 3 # dx 2 + 3 # dx 3 = ln x - 1 - 3 - 3 2 + C
2
3
x - 3 x + 3 x - 1 x - 1 ^ x - 1h ^ x - 1h x - 1 2^ x - 1h
v # dx = 1 arctanx - 5 arctan x 5 + C
4
2
x + 6 x + 5 4 20 5
1. Factor: = # dx .
2
2
_ x + 5 _i x + 1i
1 Ax + B Cx + D
2. Write the partial fractions: = + .
2
2
2
2
_ x + 5 _i x + 1i x + 5 x + 1
2 2
B x + i
3. Multiply by LCD: 1 = ^ Ax + h _ 1 + ^ Cx + D _ h x + 5i.
4. Plug in the easiest numbers to work with, 0 and 1, to effortlessly get two equations.
x = : 0 1 = B + 5 D
x = : 1 1 = 2 A + 2 B + 6 C + 6 D
5. After FOILing out the equation in Step 3, equate coefficients of like terms to come up with two
more equations.
2
The x term gives you 0 = B + D
1 1
This equation plus the first one in Step 4 give you B = - , D =
4 4
3
The x term gives you 0 = A + C
Now this equation plus the second one in Step 4 plus the known values of B and D give you
A = 0 and C = 0.
6. Split up and integrate.
1 1
- dx dx
# dx = # 4 + # 4
2
4
2
2
x + 6 x + 5 x + 5 x + 1
= - # dx + # dx
1
1
4 x + 5 4 x + 1
2
2
1 x 1
= - arctan + arctan x + C
4 5 5 4