Match the rational functions with the correct form of the partial fractions.
[\( p,q,r,a,b,c,A,B,C \) are arbitrary constants]
Match List-I with List-II
| List-I | List-II | ||
| [Rational Function] | [Partial Fraction] | ||
| (A) | \( \displaystyle\frac{px^{2}+qx+r}{(x-a)(x^{2}+bx+c)} \) | (I) | \( \displaystyle\frac{A}{x-a}+\frac{B}{(x-a)^{2}}+\frac{C}{x-b} \) |
| (B) | \( \displaystyle\frac{px^{2}+qx+r}{(x-a)^{2}(x-b)},a\ne b \) | (II) | \( \displaystyle\frac{A}{x-a}+\frac{B}{(x-a)^{2}} \) |
| (C) | \( \displaystyle\frac{px^{2}+qx+r}{(x-a)(x-b)(x-c)},a\ne b\ne c \) | (III) | \( \displaystyle\frac{A}{x-a}+\frac{Bx+C}{x^{2}+bx+c} \) |
| (D) | \( \displaystyle\frac{px+q}{(x-a)^{2}} \) | (IV) | \( \displaystyle\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (III), (B) - (I), (C) - (IV), (D) - (II)
The form of the partial fractions depends on the factors of the denominator: a linear factor (x − a) gives \( \frac{A}{x-a} \); a repeated factor \( (x-a)^{2} \) gives \( \frac{A}{x-a}+\frac{B}{(x-a)^{2}} \); an irreducible quadratic gives \( \frac{Bx+C}{x^{2}+bx+c} \).
(A) \( \frac{px^{2}+qx+r}{(x-a)(x^{2}+bx+c)} \)
This matches (III) \( \frac{A}{x-a}+\frac{Bx+C}{x^{2}+bx+c} \): one linear factor and one quadratic factor.
(B) \( \frac{px^{2}+qx+r}{(x-a)^{2}(x-b)},a\ne b \)
This matches (I) \( \frac{A}{x-a}+\frac{B}{(x-a)^{2}}+\frac{C}{x-b} \): a repeated linear factor and a distinct linear factor.
(C) \( \frac{px^{2}+qx+r}{(x-a)(x-b)(x-c)},a\ne b\ne c \)
This matches (IV) \( \frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c} \): three distinct linear factors.
(D) \( \frac{px+q}{(x-a)^{2}} \)
This matches (II) \( \frac{A}{x-a}+\frac{B}{(x-a)^{2}} \): a repeated linear factor only.
So the matching is (A)-(III), (B)-(I), (C)-(IV), (D)-(II).
NCERT (2026-27 edition):
Integration by Partial Fractions
