Page 232 - Calculus Workbook For Dummies

216 Part IV: Integration and Infinite Series

3
2
*w # 4 x + 3 x + 2 x + 1 dx = 1 ln _ 9 x + 1i x - 1 5 x + 1 + arctanx + C
2
C
4
x - 1 2
2
3
1. Factor: = # 4 x + 3 x + 2 x + 1 dx.
2
^ x - 1 ^h x + h x + 1i
1 _
3
2
4 x + 3 x + 2 x + 1 A B Cx + D
2. Write the partial fractions: 2 = + + 2 .
^ x - 1 ^h x + h x + 1i x - 1 x + 1 x + 1
1 _
3. Multiply by the LCD:
2
2
2
3
1
4 x + 3 x + 2 x + = A x + h x + i B x - h x + i Cx + D x - 1 ^h x + 1h
^
1 + ^
^ h
1 +
^
1 _
1 _
4. Plug in roots.
x = : 1 10 = 4 A A = . 2 5
x = - : 1 - 2 = - 4 B B = . 0 5
3
5. Equating the coefficients of the x term gives you C.
4 = A + B + C
5
A 2 . , B = . , so C = 1
=
5
6. Plugging in zero and the known values of A, B, and C gets you D.
1 = . 2 5 - . 0 5 - D
D 1
=
7. Integrate.
3
2
# 4 x + 3 x + 2 x + 1 dx = . 2 5 # dx + .5 # dx + # x + 1 dx
4
2
x - 1 x - 1 x + 1 x + 1
2
5
5
. ln x -
= 2 5 1 + . ln x + 1 + . ln x + 1 + arctan x + C
1 5
2
= ln _ 9 x + 1i x - 1 x + 1 + arctan x + C
C
2
2
2
*x # x - x dx = 1 ln ^ x + 1h - arctanx + 2 2 arctan x 2 + C
2
2
2
^ x + h x + 1 _i x + 2i 6 x + 2 3 2
1 _
1. Break the already factored function into partial fractions.
2
x - x = A + Bx + C + Dx + E
2
2
2
2
1 _
^ x + h x + 1 _i x + 2i x + 1 x + 1 x + 2
2. Multiply by LCD.
2
2
2
2
2
x
x - = A x + 1 _i x + i Bx + C ^h x + h x + i Dx + E ^h x + h x + 1i
1 _
_
2 + ^
1 _
2 + ^
3. Plug in single root (–1).
1
2 = 6 A A =
3
1
4. Plug 0, 1, and –2 into x and into A.
3
2
x = : 0 0 = 3 + 2 C + E
2
x = : 1 0 = + 6 B + 6 C + 4 D + 4 E
x = : 2 6 = 10 + 12 B - 6 C + 10 D - 5 E
4
5. Equate coefficients of the x terms (with A = 1 ): 0 = 1 + B + D.
3 3