Integration by parts — derivation from product rule, LIATE mnemonic

4.2.7 · Maths › Calculus II — Integration

YEH rule exist hi kyun karta hai?

RULE kya hai? (derive karne se pehle bataya gaya)

DERIVE kaise karte hain? (scratch se — derivation-from-scratch)

Formula Step-by-step derivation

Step 1 — Product rule se shuru karo. Do functions $u (x)$ aur $v (x)$ ke liye: $\frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x}$ Yeh step kyun? Yahi ek cheez hum assume karte hain — yeh ek jaana-maana, trusted identity hai.

Step 2 — Dono sides ko $x$ ke respect mein integrate karo. $\int \frac{d}{d x} (uv) d x = \int u \frac{d v}{d x} d x + \int v \frac{d u}{d x} d x$ Yeh step kyun? Integration, differentiation ko undo karta hai, toh messy products aise integrals ban jaate hain jinhein hum rearrange kar sakte hain.

Step 3 — Left side simplify karo. Kyunki integration derivative ko undo karta hai: $uv = \int u \frac{d v}{d x} d x + \int v \frac{d u}{d x} d x$ Yeh step kyun? $\int \frac{d}{d x} (kuch bhi) d x = wahi kuch bhi$ (Fundamental Theorem of Calculus).

Step 4 — Jo integral chahiye uske liye solve karo. Ek term doosri taraf le jao: $\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x$ Yeh step kyun? Hum chahte hain ki $\int u d v$ left side par akela ho; doosra piece naya (aasaan) integral ban jaata hai.

Step 5 — Differential form mein likho ( $d v = \frac{d v}{d x} d x$ , $d u = \frac{d u}{d x} d x$ ): $\int u d v = uv - \int v d u$

Bas itna hi — kuch memorize nahi, sab kuch banaya gaya.

Figure — Integration by parts — derivation from product rule, LIATE mnemonic

Choice problem — $u$ kaun sa factor hai?

Worked Examples

Common Mistakes (Steel-man them)

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho tumhare paas do cheezein mili hui hain, jaise caramel ( $x$ ) aur chocolate ( $e^{x}$ ) se bani ek chipchipa candy, jo tumhe clean karni hai. Tum unhe normal melting (substitution) se alag nahi kar sakte. Lekin tum jaante ho aisi candy banani kaise hai chipkake (woh product rule hai). Toh tum gluing machine ko ulta chalao: apni messy candy ko thodi cleaner candy se trade karo, yeh trade jitni baar zaroorat ho karo, aur aakhir mein tum itni simple candy ke saath reh jaate ho ki woh sirf ek plain piece hai jise tum pehle se khaana (integrate karna) jaante ho. LIATE sirf "pehle kaun sa piece chheelun" ki instructions hain taaki tum hamesha sahi side cheelo.

Active-Recall Flashcards

Integration by parts ka formula kya hai?

\int u d v = uv - \int v d u

Integration by parts kis identity se derive hota hai?

Product rule

\frac{d}{d x} (uv) = u v^{'} + v u^{'}

se, dono sides ko integrate karke.

LIATE ka full form kya hai?

Logarithmic, Inverse-trig, Algebraic, Trigonometric, Exponential —

u

choose karne ka order.

LIATE mein kaun sa function

u

banta hai?

Jo list mein pehle aata hai (use differentiate kiya jaata hai).

u

ko woh function kyun chunte hain jo differentiation par simplify ho?

Kyunki rule

\int u d v

\int v d u

se replace karta hai; yeh tab hi helpful hota hai jab

d u

integral ko aasaan banaaye.

\int ln x d x

kaise integrate karte hain?

ln x \cdot 1

likho,

u = ln x, d v = d x

lo →

x ln x - x + C

\int x e^{x} d x = ?

e^{x} (x - 1) + C

"Loop"/algebraic trick kab use karte hain?

Jab repeated parts ke baad original integral wapas aaye (jaise

\int e^{x} sin x d x

); algebraically solve karo.

\int e^{x} sin x d x = ?

\frac{1}{2} e^{x} (sin x - cos x) + C

d v

mein

d x

kyun hona chahiye?

Kyunki

v = \int d v

;

d x

mark karta hai ki kya integrate ho raha hai.

Connections

Product rule (differentiation) — source identity.
Fundamental Theorem of Calculus — justify karta hai ki $\int \frac{d}{d x} (uv) d x = uv$ .
Integration by substitution — doosri main technique; pehle ise try karo.
Reduction formulae — parts baar baar apply karke bante hain.
Tabular integration (DI method) — polynomial × (exp/trig) ke liye shortcut.
Definite integrals — limits ke saath parts: $[uv]_{a}^{b} - \int_{a}^{b} v d u$ .