082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

1.7 Partial fractions 45
EXERCISES1.7
1 Calculate the partial fractionsofthe following
fractions:
6x+14 7−2x
(a)
x 2 (b)
+4x+3 x 2 −x−2
3x+6 8 −x
(c)
2x 2 (d)
+3x 6x 2 −x−1
(e)
13x 2 +11x +2
(x +1)(2x +1)(3x +1)
2 Calculatethe partialfractionsofthe following
fractions:
2x+7 4x−5
(a)
x 2 (b)
+6x+9 x 2 −2x+1
(c)
(d)
3x 2 +8x+6
(x 2 +2x +1)(x +2)
3x 2 −3x−2
(x 2 −1)(x −1)
(e)
3x 2 +7x+6
x 3 +2x 2
3 Expressthe following aspartialfractions:
(a)
x 2 +x+2
(x 2 +1)(x +1)
(b)
(c)
(d)
(e)
5x 2 +11x+5
(2x +3)(x 2 +5x +5)
4x 2 +5
(x 2 +1)(x 2 +2)
18x 2 +7x +44
(2x −3)(2x 2 +5x +7)
2x
(x 2 −x+1)(x 2 +x+1)
4 Express the following fractions aspartialfractions:
(a)
(c)
(d)
(e)
x 2 +7x+13
x +4
x 2 +8x+2
x 2 +6x+1
x 3 −2x 2 +3x−3
x 2 −2x+1
2x 3 +2x 2 −2x−1
x 2 +x
(b)
12x−4
2x−1
Solutions
1 (a)
(c)
(e)
2 (a)
(c)
(d)
(e)
3 (a)
2
x +3 + 4
x +1
2
x − 1
2x+3
(b)
(d)
2
x +1 + 1
2x+1 − 1
3x+1
2
x +3 + 1
(x +3) 2 (b)
1
x +1 + 1
(x +1) 2 + 2
x +2
2
x −1 − 1
(x −1) 2 + 1
x +1
2
x + 3 x 2 + 1
x +2
1
x +1 + 1
x 2 +1
1
x −2 − 3
x +1
3
2x−1 − 5
3x+1
4
x −1 − 1
(x −1) 2
(b)
(c)
(d)
(e)
2x
x 2 +5x+5 + 1
2x+3
1
x 2 +1 + 3
x 2 +2
5
2x−3 + 4x−3
2x 2 +5x+7
1
x 2 −x+1 − 1
x 2 +x+1
4 (a) x+3+ 1
x +4
(c) 1+ 2x+1
x 2 +6x+1
(d) x+ 2
x −1 − 1
(x −1) 2
(e) 2x − 1 x − 1
x +1
(b) 6+ 2
2x−1