When decomposing $\dfrac{px^2+qx+r}{(x-a)^2(x-b)}$ into partial fractions, the correct form is
A$\dfrac{A}{x-a}+\dfrac{B}{x-b}+\dfrac{C}{(x-b)^2}$
B$\dfrac{A}{x-a}+\dfrac{Bx+C}{(x-a)^2}+\dfrac{D}{x-b}$
C$\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}+\dfrac{C}{x-b}$
D$\dfrac{Ax+B}{(x-a)^2}+\dfrac{Cx+D}{x-b}$
Answer & Solution
Correct answer: C. $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}+\dfrac{C}{x-b}$
1. A repeated linear factor $(x-a)^2$ contributes two terms: $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}$.
2. The distinct factor $(x-b)$ contributes one term $\dfrac{C}{x-b}$.
3. Option A wrongly repeats $(x-b)$ and ignores the repetition of $(x-a)$.
4. Options B and D wrongly place linear numerators over a repeated linear factor; the form is $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}+\dfrac{C}{x-b}$ (Table 7.2 case 4).
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.316_
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