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

4.2.6 · D2 · Maths › Calculus II — Integration › U-substitution — technique, change of limits for definite in

Step 1 — Integral sign aakhir maang kya raha hai

KYA. Hum ek curve ke neeche patli rectangles stack kar rahe hain aur unki areas add kar rahe hain.

KYUN. U-substitution ka har symbol in slivers ki manipulation hai. Agar hum yeh nahi jaante ki $d x$ ek width hai — ek real geometric length — toh phrase " $d u = g^{'} (x) d x$ " ka koi matlab nahi. Isliye hum ise yahan pehle anchor karte hain.

PICTURE. Ek amber strip dekho: uski top curve ko height $h (x)$ par touch karti hai, uska base width $d x$ ka hai. Poora shaded region hazaron aisi strips ka sum hai.

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

Step 2 — Ek hi road ke liye do rulers: $u = g (x)$ introduce karna

Ab hum bottom axis ke saath ek doosra measuring tape lagate hain.

KYA. Humne straight ruler ( $x$ ) ke neeche ek curved doosra ruler ( $u$ ) laaya. Kyunki $g$ usually straight line nahi hoti, $x$ mein equal steps $u$ mein unequal steps ban jaate hain.

KYUN yeh tool aur koi nahi? Hum rename karte hain kyunki integrand ke andar secretly $g (x)$ chhupa hua hai. Parent se poori baat yeh hai ki integration Chain Rule ko reverse karta hai: $F (g (x))$ differentiate karne se ek inner $g (x)$ aur ek outer $F^{'}$ produce hua tha. Inner function ko ek single letter $u$ mein rename karna poori gaddaddi collapse kar deta hai.

PICTURE. Dekho ki $x$ -ruler (bottom) par teen evenly spaced ticks $u$ -ruler par teen unevenly spaced ticks par map karti hain (curved arrows). Woh unevenness hi Step 3 ki poori kahani hai.

Step 3 — $d u = g^{'} (x) d x$ KYUN hai: ruler ka stretch factor

Yahan picture ka dil hai. Jab tum purane ruler par ek tiny step $d x$ lete ho, toh naye ruler par $d u$ ka kitna bada step lete ho?

KYA. Humne measure kiya ki naya ruler har point par kitna stretch ya squeeze hua hai.

KYUN. Ek definite integral areas ka sum hai. Agar hum $u$ -ruler par switch kar lein lekin slivers apni purani widths rakhi rahe, toh hum galat widths add kar rahe honge aur galat area milega. Factor $g^{'} (x)$ woh correction hai jo har sliver ki area honest rakhti hai. Yeh purani-width $d x$ ko naye-width $d u$ mein convert karti hai.

PICTURE. Do triangles same rise ko alag-alag runs ke against measure karte dikhate hain. Jahaan $g$ steep hai (badi $g^{'}$ ), ek chhota $d x$ ek bade $d u$ mein fan out hota hai — $u$ -axis par wide amber gap dekho.

Step 4 — Chain-rule reversal, drawn

Ab hum picture ko actual theorem se connect karte hain. Ek composed function ka differentiation do factors produce karta hai; hum machine ko ulta chalate hain.

KYA. Humne pehchaana ki integrand mein baitha messy factor $g^{'} (x) d x$ aur kuch nahi balki $d u$ chhupa hua hai, aur $f (g (x))$ sirf height $f (u)$ hai.

KYUN zaruri hai ki $g^{'} (x)$ present ho, koi aur factor nahi. U-substitution tabhi "click" karta hai jab integrand pehle se apna stretch factor $g^{'} (x)$ carry karta ho. Yahi chain rule ka fingerprint hai jo peeche chhhod jaata hai. Agar yeh absent hai (ya kisi variable se off hai), Step 4 fail ho jaata hai — Step 6 dekho.

PICTURE. Same shaded region jaise Step 1 mein, ab dono rulers drawn hain. Strip ko do baar label kiya gaya hai: bottom ruler par $f (g (x)) d x$ aur top ruler par $f (u) d u$ . Ek shape, do naam.

Step 5 — Limits KYUN move karni chahiye (definite integrals)

Do walls $x = a$ aur $x = b$ purane ruler par positions theen. Jab ek baar hum purana ruler phenk dete hain aur sirf $u$ padhte hain, toh un walls ko unke naye naam chahiye.

KYA. Humne dekha ki do boundary walls $u$ -ruler par kahaan land karti hain.

KYUN. Agar hum $u$ par switch karne ke baad lazily $\int_{a}^{b}$ rakh lete, toh hum naye ruler par $a$ aur $b$ label ki positions ke beech integrate kar rahe hote — bilkul alag road stretch. Area galat hota.

PICTURE. Left wall aur right wall vertical amber lines ki tarah drawn hain. Har ek ko bottom ruler ( $a, b$ ) tak follow karo aur top ruler ( $g (a), g (b)$ ) tak across. Same walls, relabelled.

Step 6 — Degenerate case: jab stretch factor vanish ho ya reverse ho jaaye

Hume har scenario cover karna hai, including woh jo naive picture ko tod dete hain.

KYA. Humne check kiya kya hota hai jab stretch factor zero ya negative ho.

KYUN. Ek reader jisne sirf increasing $g$ dekha ho woh panic kar jaata jab naye limits "ulte" nikle ( $5 \to 1$ instead of $1 \to 5$ ). Picture dikhati hai ki yeh error nahi hai — yeh orientation bookkeeping hai.

PICTURE. Left panel: $g^{'} = 0$ , teen $x$ -ticks ek $u$ -tick par collapse (crush). Right panel: $g$ decreasing, arrows cross — $x = a$ $u$ -ruler par $x = b$ ke right par land karta hai.

Step 7 — Worked example, area ke roop mein dekha gaya

PICTURE. Left: original curve $x x^{2} + 1$ , $x \in [0, 2]$ par, shaded. Right: transformed curve $\frac{1}{2} u$ , $u \in [1, 5]$ par, shaded. Do shaded areas equal hain — woh equality hi U-substitution hai. Dono par same number $3.39$ annotated hai.

Recall Peek karne se pehle predict karo

Kyunki $u = x^{2} + 1$ tezi se badhta hai, kya naye limits $0 \to 2$ se zyada door honge ya paas? ::: Zyada door — $1 \to 5$ (span 4 vs span 2), kyunki steep stretch factor $g^{'} = 2 x$ positions ko ek doosre se dur kheenchta hai.

Ek picture mein summary

Upar sab kuch ek single diagram mein: same amber region, $x$ -ruler ke neeche measure kiya gaya (left) aur $u$ -ruler ke neeche (right), walls relabelled hain aur stretch factor $g^{'} (x)$ dono widths ko connect karta hai. Ise left-to-right padho aur tumne poora theorem re-derive kar liya.

Recall Feynman retelling (poora walkthrough ek dost ko explain karo)

Socho ek road par ek normal tape measure ( $x$ ) chipki hui hai, aur ek fence ke neeche ki area woh hai jo hum chahte hain. Ab upar ek stretchy tape measure ( $u = g (x)$ ) chipkao: jahan road "busy" hai, stretchy tape apne numbers door door failaati hai; jahan calm hai, woh bunch up ho jaati hai. Jis jagah tape kitni stretch hui hai woh uski slope $g^{'} (x)$ hai — isliye ek derivative aata hai. Jab hum stretchy tape se fence ki area measure karte hain, har patli strip ka wahi real area rehta hai (ek taller-dikhne wali strip sirf narrower hoti hai compensate karne ke liye, aur vice-versa) — yahi equation $f (g (x)) g^{'} (x) d x = f (u) d u$ hai. Aakhir mein, fence ke do end-posts ek inch bhi nahi hile, lekin stretchy tape par ab woh alag numbers $g (a)$ aur $g (b)$ read karte hain — isliye hum limits change karte hain. Agar stretchy tape kabhi ulti chale ( $g^{'} < 0$ ), toh do posts order swap kar lete hain, aur built-in minus sign automatically answer fix kar deta hai.

Connections

Chain Rule — fingerprint $g^{'} (x)$ integrand ke andar yahan se aata hai; U-sub ise reverse karta hai.
Definite Integral & Fundamental Theorem of Calculus — Step 1 mein "area = sum of slivers" picture.
Antiderivatives & Indefinite Integrals — integral $u$ mein aane ke baad hum yahi compute karte hain.
Integration by Parts — product-rule ka inverse; geometry se contrast karo.
Trigonometric Substitution — $sin$ ya $tan$ ki shape ka ek stretchy ruler.
Hinglish version