Page 34 - DMTH202_BASIC_MATHEMATICS_II

Unit 2: Integration by Partial Fraction

Notes
dx dx
2

3


 3   2 (x  2) dx  6 (x  2) dx  3 

x  2 x  1
2 3
 3log(x  2)  
(x  2) (x  2) 2
x  2 2x  7
 3log 
x  1 (x  2) 2
Self Assessment
Fill in the blanks:
11. Every proper rational fraction can be expressed as a sum of ............................ fractions.
12. To each linear factor ax+b occurring once in the denominator of a proper rational fraction,
A
there corresponds a ............................ partial fraction of the form , where A is a

ax b
constant to the determined.
13. In case of ............................ Linear Factors, to each linear factors ax+b occurring n times in
the denominator of a proper rational fraction.

14. In case of Distinct Quadratic Factors, to each ............................ quadratic factor occurring
once in the denominator of a proper rational fraction, there corresponds a simple partial

Ax B
fraction of the form 2 where A and b are constants to be determined.
9x  bx c

15. In case of Repeated Quadratic Factors, to each irreducible quadratic factor, occurring n
times in the denominator of a proper rational fraction, there corresponds a ............................
of n partial fractions.

2.4 Summary

 By the method of “partial fractions” we can translate any rational function into a polynomial
and fractions each one with negative powers of just one factor (x–a).
 Each polynomial can be factored into linear factors (if complex numbers can occur in the
factors).
 The partial fraction theorem permits us to integrate P(X) / Q(x); we are required only to
integrate the polynomial D(x) and the different inverse powers appearing in this sum.

 In partial fraction theorem, by substraction of a suitable multiple of an suitable inverse
power, we can attain a rational function that is less singular than R(x) / Q(x) at an random
root of Q.
 Finding the Coefficients in the Partial Fraction Expansion includes four methods such as
expansion, cover up, evaluate and solve equations, and common denominator.

 Every proper rational fraction cab be expressed as a sum of simple fractions whose
n
2
denominators are of the form (9x+b) and (9x +bx+c) , n being a positive integer.
n
 To each linear factor ax+b occurring once in the denominator of a proper rational fraction,
A
there corresponds a single partial fraction of the form , where A is a constant to the
ax b

determined.

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