Mathematical Formulas for Industrial and Mechanical Engineering

Case III. Distinct Quadratic Factors

To each irreducible quadratic factor ax

2

1 bx 1 c occurring once in the denomi-

nator of a proper rational fraction, there corresponds a single partial fraction of the

form ðAx 1 BÞ=ðax

2

1 bx 1 cÞ, where A and B are const ants to be determined.

Case IV. Repeated Quadratic Factors

To each irreducible quadratic factor ax

2

1 bx 1 c occurring n times in the denomi-

nator of a proper rational fraction, there corresponds a sum of n partial fractions of

the form

A

1

x 1 B

1

ax

2

1 bx 1 c

1

A

2

x 1 B

2

ðax

2

1bx1cÞ

2

1 ? 1

A

n

x 1 B

n

ðax

2

1bx1cÞ

n

where the As and Bs are constants to be determined.

5.19 Properties of Trigonometric Functions

Properties ...