Page 215 - Calculus Demystified

CHAPTER 7
202
7.2 Partial Fractions Methods of Integration
7.2.1 INTRODUCTORY REMARKS

The method of partial fractions is used to integrate rational functions, or quotients
of polynomials. We shall treat here some of the basic aspects of the technique.
The first fundamental observation is that there are some elementary rational
functions whose integrals we already know.

I Integrals of Reciprocals of Linear Functions An integral

1
dx
ax + b

with a = 0 is always a logarithmic function. In fact we can calculate

1 1 1 1
dx = dx = log |x + b/a|.
ax + b a x + b/a a

II Integrals of Reciprocals of Quadratic Expressions An integral

1
dx,
c + ax 2
when a and c are positive, is an inverse trigonometric function. In fact we can use
what we learned in Section 6.6.3 to write

1 1 1
dx = dx
c + ax 2 c 1 + (a/c)x 2

1 1
= √ dx
c 1 + ( a/cx) 2
√
1 a 1
= √ √ √ dx
ac c 1 + ( a/cx) 2
1
= √ arctan a/cx + C.
ac
III More Integrals of Reciprocals of Quadratic Expressions An integral

1
dx
2
ax + bx + c