Page 35 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes 2

 To each irreducible quadratic factor ax  bx c occurring once in the denominator of a

Ax B
proper rational fraction, there corresponds a simple partial fraction of the form 2
9x  bx c

where A and b are constants to be determined.
2.5 Keywords

2
Distinct Quadratic Factor: To each irreducible quadratic factor ax  bx c occurring once in

the denominator of a proper rational fraction, there corresponds a simple partial fraction of the
Ax B

form 2 where A and b are constants to be determined.

9x  bx c
Partial Fraction: 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).
2.6 Review Questions

1. What is partial fraction theorem? Illustrate the proof of partial fraction theorem.
2. Depict various methods used in finding the Coefficients of the Partial Fraction Expansion.
3. Explicate the working of Method 3: evaluate and solve equations. Give example.
4. Make distinction between Linear Quadratic Factors and Repeated Quadratic Factors.

3x
5. Write the fractions as sum of partial fractions and then integrate with respect
(x  2)x  4)
to x.

1
6. Write the fractions 2 as sum of partial fractions and then integrate with respect
x  4x  4
to x.

x 2
7. Write the fractions as sum of partial fractions and then integrate with
(x  1)(x  2)(x  3)
respect to x.

x
8. Write the fractions as sum of partial fractions and then integrate with
(x  1)(3x  2)(x  3)
respect to x.

1
9. Write the fractions as sum of partial fractions and then integrate with respect to x.
( x x  2)

1
10. Write the fractions 3 as sum of partial fractions and then integrate with respect to x.
x  x
Answers: Self Assessment

1. partial fractions 2. rational
3. proper 4. long division

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