Partial fractions — linear, repeated, irreducible quadratic factors

4.2.10 · Maths › Calculus II — Integration

Kya / Kyun / Kaise

KAISE (master recipe):

Use proper banao (zarurat ho toh long division karo).
$Q (x)$ ko reals ke upar linear aur irreducible quadratic factors mein fully factor karo.
Har factor ke liye, unknown numerators ka sahi template likho (niche ke rules).
Unknowns solve karo (cover-up / matching coefficients).
Har simple piece integrate karo.

Formula Teen templates (iska dil)

$Q$ ke har distinct factor ke liye, matching block add karo:

$Q$ mein Factor	Jo block add karte hain
linear $(x - a)$	$\frac{A}{x - a}$
repeated linear $(x - a)^{k}$	$\frac{A _{1}}{x - a} + \frac{A _{2}}{( x - a ) ^{2}} + \dots + \frac{A _{k}}{( x - a ) ^{k}}$
irreducible quadratic $(x^{2} + b x + c)$	$\frac{B x + C}{x ^{2} + b x + c}$
repeated quadratic $(x^{2} + b x + c)^{m}$	$\sum_{j = 1}^{m} \frac{B _{j} x + C _{j}}{( x ^{2} + b x + c ) ^{j}}$

Counting check: unknowns ki total sankhya $= de g Q$ . Hamesha.

Cover-up rule ki derivation (first principles)

Ek simple linear factor lo. Likho $\frac{P ( x )}{( x - a ) g ( x )} = \frac{A}{x - a} + \frac{(rest)}{g ( x )} .$ Dono sides ko $(x - a)$ se multiply karo: $\frac{P ( x )}{g ( x )} = A + (x - a) \cdot \frac{(rest)}{g ( x )} .$ Ab $x = a$ set karo. Doosre term mein $(x - a) \to 0$ ka factor hai, isliye woh vanish ho jaata hai: $A = \frac{P ( a )}{g ( a )}$ yaani denominator mein $(x - a)$ ko cover up karo aur jo bacha usmein $x = a$ plug karo. Yeh bas ek aisa limit evaluate karna hai jo baaki sab terms ko khatam kar deta hai. (Cover-up highest-power coefficient cleanly deta hai; lower repeated/quadratic coefficients ke liye abhi bhi matching chahiye.)

Worked examples

Figure — Partial fractions — linear, repeated, irreducible quadratic factors

Quadratic piece integrate karna (woh part jisse log darate hain)

$\frac{B x + C}{x ^{2} + b x + c}$ ke liye (irreducible), do ideas mein split karo:

Log part: numerator ko denominator ke derivative $(2 x + b)$ ke proportional banana → $\frac{B}{2} ln (x^{2} + b x + c)$ deta hai.
Arctan part: bacha hua constant, completing the square $x^{2} + b x + c = (x + \frac{b}{2})^{2} + d^{2}$ ke baad, $\frac{1}{d} arctan \frac{x + b /2}{d}$ deta hai.

Common mistakes (steel-manned)

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho kisi ne teen simple juices ek bade cup mein mila diye aur tumhe recipe pata karni hai. Partial fractions woh kaam hai jisme tum mix taste karke bolte ho "ah, itna apple, itna orange, itna grape." Bada messy fraction woh mixed cup hai; simple fractions $\frac{A}{x - a}$ pure juices hain. Jab recipe pata chal jaaye, har pure juice "peena" (integrate karna) aasaan hai — seedhe waale $ln$ bante hain, curvy quadratic waale $ln$ plus ek $arctan$ swirl dete hain.

Active recall

Partial fractions seedha kab apply kar sakte ho?

Sirf jab proper ho,

de g P < de g Q

; warna pehle long-divide karo.

Decomposition mein unknown constants ki total sankhya kitni hoti hai?

de g Q

(denominator ki degree).

Repeated linear factor

(x - a)^{3}

ka template?

\frac{A _{1}}{x - a} + \frac{A _{2}}{( x - a ) ^{2}} + \frac{A _{3}}{( x - a ) ^{3}}

Irreducible quadratic factor

(x^{2} + b x + c)

ka template?

\frac{B x + C}{x ^{2} + b x + c}

(linear numerator).

\frac{P}{( x - a ) g}

ke liye cover-up rule batao.

A = \frac{P ( a )}{g ( a )}

(x - a)

cover karo,

x = a

set karo.

Cover-up kyun kaam karta hai?

(x - a)

se multiply karo aur

x = a

set karo; baaki har term mein

(x - a) \to 0

ka factor rehta hai.

Kaise check karte hain ki ek quadratic irreducible hai?

Discriminant

b^{2} - 4 c < 0

(koi real roots nahi).

\int \frac{A}{x - a} d x = ?

A ln ∣ x - a ∣ + C

\int \frac{A}{( x - a ) ^{2}} d x = ?

- \frac{A}{x - a} + C

(power rule, log nahi).

\frac{B x + C}{x ^{2} + b x + c}

integrate karne ki strategy?

Numerator ko

(2 x + b)

ke proportional banao log part ke liye; arctan part ke liye complete the square karo.

Quadratic factor ko

B x + C

kyun chahiye, sirf

A

nahi?

Numerator degree denominator se ek kam honi chahiye taaki saare proper fractions span ho sakein.

Connections

Polynomial long division — functions ko proper banana ke liye prerequisite.
Integration by substitution — $\frac{1}{2} ln (x^{2} + \dots)$ step ke liye use hota hai.
Completing the square — quadratics ko arctan form mein convert karta hai.
Standard integrals — log and arctan — $\int \frac{d x}{x ^{2} + a ^{2}} = \frac{1}{a} arctan \frac{x}{a}$ .
Laplace transforms — inverse via partial fractions — wahi algebra reuse hoti hai.
Factoring polynomials over the reals — factor list provide karta hai.