Calculus: Analytic Geometry and Calculus, with Vectors

9.4 Partial fractions 511

9 Supposing that a and b are constants for which b'- a'-, obtain the formula

f ea° cosh ht dt =eat(b sinh bt - a cosh bt)

b2-a2 +c

by integrating by parts twice. For the case where b2 = a2 see the next problem.

(^10) Considering separately the three cases in which b2 74 a2, b= a, and b =
-a, evaluate the integral

f [e(a+b)e+ e(a-b) 1J dt

and put the answers in terms of hyperbolic functions.

11 Determine whether

(a) lim fix to dt x t-1 dt

a-.-1 1

(b) lim fox eal cosh bt di =fox ea0 cosh at dt

b- a

9.4 Partial fractions

(9.411) f(x) =x2-2x+1+x2 1+x'+1+x2I

.

Use of basic integration formulas then gives

(9.412) (x2+1)

+tan-' x+c.

Adding the terms in the right member of (9.411) gives

(9.413) f(x) =xb - 3x4 + 4x3 - x2 + 3x

x3-x2+x-1

and shows us that the integral

(9.414) f xs - 3x4 + 4x3 - x22 + 3xdx

J xs-x2+x-1

is equal to the right member of (9.412). Interest in this business starts

to develop when we wonder how we would evaluate the integral (9.414)
if it were handed to us without the preceding formulas. The answer to

this question is quite straightforward. We learn and use a procedure
by which the preceding formulas can be worked out.
The first step is to look at (9.414). The integrand is a quotient of
polynomials in x, and the degree of the numerator is not less than the
degree of the denominator. In such cases we employ division (or long
division) to obtain a polynomial and a new quotient in which the degree