Calculus: Analytic Geometry and Calculus, with Vectors

9.4 Partial fractions 515

2 Supposing that p, q, r are three different constants, determine A, B, C

such that

1 1 B + C

(x-p)(x-q)(x-r) x - p x -q x-r

and use the result to obtain the integral with respect to x of the left member.
Ans.:

(pq)(p-r)loglx-pI +(q-1p) (q-r)logIx-qj

+( p)(1

r-q)

r - logy - rj+c.

3 Supposing that p q, determine 11, B, C such that

1 __ R + B + C

(x - p)2(x - q) x - p (x - p)2 x - q

and use the result to obtain the integral with respect to x of the left member.
Ins.:

1

1 1

+^1 2logIx-qj +c.

(q-p) q - px - p U=-p-)

4 Obtain the answer to Problem 3 by starting with the tricky calculation

1 _ 1 q-p _ 1 (x-p)-(x-q)

(x - p)2(x - q) q - p (x - p)2(x - q) q - p (x - p)2(x - q)

1 1 _ (^11)
q - p L (x-p)(x -q) (x-p)2
5 Assuming that p, q, and r are different constants, find the partial fraction
expansions of the following quotients and check your answers.
x3 1
(a) x2 - 4 (b)x(x2 - 4)
(c) x (,A x
(x - 1) (x - 2) (x - p)(x - q)
x
(e) (x - 1) (x
x
2) (x - 3)
(f) (x - p) (x
q) (x - r)
(g) (x - 1)(x
x2
2)(x - 3) (h) (x- p)(x
x?
q)(x - r)
(2) (x - 1) (x - 2)2 (9)(x - p)(x - q)2
(^6) Show that
(^7) Show that
(1)
1 dt=log2-2

It t(1 + t)2

1 m(x2+1)(x2+4)dx=6