U-substitution — technique, change of limits for definite integrals

4.2.6 · Maths › Calculus II — Integration

WHAT it is

WHY the formula is true (derivation from scratch)

Hum rule yaad nahi karte — hum isko build karte hain.

Shuru karte hain chain rule se: $\frac{d}{d x} F (g (x)) = F^{'} (g (x)) \cdot g^{'} (x) .$

Dono sides ko $x$ ke respect mein integrate karo. Left side ek derivative ka antiderivative hai, toh woh original function return karta hai: $\int \frac{d}{d x} F (g (x)) d x = F (g (x)) + C .$

Isliye: $\int F^{'} (g (x)) g^{'} (x) d x = F (g (x)) + C .$

Ab rename karo: maano $u = g (x)$ aur $f = F^{'}$ (toh $F$ , $f$ ka ek antiderivative hai). Right side hai $F (u) + C$ , jo exactly $\int f (u) d u$ hai. Hence: $\int f (g (x)) g^{'} (x) d x = \int f (u) d u$

HOW to do it — the recipe

Choose karo $u = g (x)$ : usually inner function, yaani woh cheez jo kisi power/root/exp/log/trig ke andar hai, YA woh function jiska derivative bhi present ho.
Differentiate karo: $d u = g^{'} (x) d x$ compute karo.
Replace karo har $x$ -piece ( $d x$ sameit) taaki integral poora $u$ mein ho jaaye.
$u$ mein Integrate karo.
Indefinite: $u = g (x)$ back substitute karo. Definite: limits change karo (neeche dekho) aur kabhi back substitute mat karo.

Figure — U-substitution — technique, change of limits for definite integrals

Worked examples

Steel-manned mistakes

Flashcards

U-substitution kaunse differentiation rule ko reverse karta hai?

Chain rule ko.

Agar

u = g (x)

ho, toh

d u

kya hai?

d u = g^{'} (x) d x

General u-sub identity kya hai?

\int f (g (x)) g^{'} (x) d x = \int f (u) d u

DEFINITE integral mein substitute karne ke baad limits ka kya hota hai?

Woh

u

-values mein change ho jaati hain:

\int_{g (a)}^{g (b)} f (u) d u

Aap back-substitution AUR limits change ek saath kyun nahi kar sakte?

Yeh dono equivalent finishing methods hain; dono karna endpoints ko double-translate kar deta hai aur galat answer deta hai.

Tumhare paas

x d x

hai lekin

d u = 2 x d x

hai — tum kya likhoge?

x d x = \frac{1}{2} d u

(sirf constant fix karo).

Kya aap integral se bahar ek stray variable jaise

1/ x

nikaal sakte ho?

Nahi — sirf constants bahar ja sakte hain; balki

u = g (x)

x

ke liye solve karo.

\int_{0}^{2} x x^{2} + 1 d x

ke liye

u = x^{2} + 1

ke saath nayi limits kya hain?

u : 1 \to 5

U-sub recipe ka pehla step kya hai?

u = g (x)

choose karo — usually inner function jiska derivative present ho.

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

Socho ek lamba messy word jo actually kisi simple cheez ka nickname hai. U-substitution mein hum messy inside part ko ek chota nickname "u" dete hain. Achanak poori problem choti aur easy ho jaati hai. Rule " $d u =$ (u ki slope) $\times d x$ " iska tarika hai ki jab hum naam badle, cheez ki quantity same rahe. Aur agar problem mein likha tha "purane naam pe point 0 se point 2 tak," toh humein figure out karna hoga ki woh dono points naye naam mein kya kehlaate hain — isliye limits change hoti hain.

Connections

Chain Rule — u-sub iska exact inverse hai.
Antiderivatives & Indefinite Integrals
Definite Integral & Fundamental Theorem of Calculus — limit-changing justify karta hai.
Integration by Parts — product-rule ka inverse (chain-rule inverse se contrast karo).
Trigonometric Substitution — ek specialized u-sub hai.