Basic integration rules — power, trig, exponential, log: Visual walkthrough

4.2.2 · D2 · Maths › Calculus II — Integration › Basic integration rules — power, trig, exponential, log

Hum in words ko scratch se build karenge, is order mein: area under a curve → slope of a curve → "undoing" a slope → raising a power → the lost constant $+ C$ .

Step 1 — Slope kya hoti hai, ek picture mein?

KYA HAI. Curve $y = x^{2}$ (ek smooth valley) dekho. Usmein koi ek point chuno aur ek ruler rakho jo wahin se curve ko sirf chhoota ho — woh ruler tangent line hai, aur uska tilt hi slope hai.

PEHLE YEH KYUN. Integration, slopes lene ka reverse hai. Ek aisi machine ko reverse nahi kar sakte jo humne kabhi forward chalte nahi dekhi. Isliye pehle slopes dekhte hain.

PICTURE. Figure mein, burnt-orange curve $y = x^{2}$ hai. Teen points par humne teal tangent rulers draw kiye hain. Dhyan do: left par ruler neeche tilt hai (negative slope), middle mein flat hai (slope $0$ ), right par upar tilt hai (positive slope). Slope badal'ti rehti hai jab aap along chalt'e ho.

Figure — Basic integration rules — power, trig, exponential, log

Woh rule jo $x^{2}$ ki slope har point par batata hai, uska derivative hai, likha jaata hai $\frac{d}{d x}$ se: $\frac{d}{d x} x^{2} = old power 2 power dropped by 1 x .$

Yahan $\frac{d}{d x}$ ka matlab hai "slope-finding machine jo aage jo likha hai usse act kar rahi hai," $2$ woh purana exponent hai jo neeche se front pe aaya, aur $x^{1}$ matlab exponent $2$ ek se ghata. Yeh kyun hota hai, dekhne ke liye Derivatives — basic rules dekhein.

Step 2 — Har derivative mein chhupa pattern

KYA HAI. Kai powers par test karo aur stack karo: $\frac{d}{d x} x^{5} = 5 x^{4}, \frac{d}{d x} x^{3} = 3 x^{2}, \frac{d}{d x} x^{1} = 1 \cdot x^{0} = 1.$

$x^{5} \to$ woh $5$ front pe jump karta hai, exponent $5 - 1 = 4$ ban jaata hai.
$x^{3} \to$ woh $3$ front pe jump karta hai, exponent $2$ ban jaata hai.
$x^{1} \to$ woh $1$ front pe jump karta hai, exponent $0$ ban jaata hai, aur $x^{0} = 1$ (koi bhi nonzero number ki power $0$ hoti hai $1$ ).

KYUN. Agar hum machine ke do moves clearly forward jaate dekh sakte hain, toh hum plan kar sakte hain ki unhe undo kaise karein peeche jaate waqt. "Power ko 1 se ghatao" ko undo karna matlab hoga "power ko 1 se badha." "Purani power se multiply karo" ko undo karna matlab hoga "kisi cheez se divide karo."

PICTURE. Staircase figure: har orange tile ek power $x^{n}$ hai; usse nikalti teal arrow (labelled "differentiate") ek step neeche land karti hai aur multiplier arrow par likhti hai. Arrows ko ulta padho (plum arrows) aur aap integrate kar rahe ho.

Step 3 — Antiderivative guess karna

KYA HAI. Hum ek aisi function chahte hain jiska slope exactly $x^{n}$ ho. Step 2 se, differentiation power ko ghatata hai, toh $x^{n}$ par land karne ke liye hume ek power upar se start karna hoga: guess karo $F = x^{n + 1}$ .

YEH GUESS KYUN. Yeh wahi guess hai jo Step 2 ke "power drops by 1" move ke baad bachti hai: $n + 1$ se shuru karo, ek se ghata, $n$ par land karo.

PICTURE. Guess ko differentiate kar ke test karo: $\frac{d}{d x} x^{n + 1} = old power to front (n + 1) x^{(n + 1) - 1} = (n + 1) x^{n} .$

$n + 1$ woh exponent hai jo humne chuna; yeh multiplier ban ke front par slide kar jaata hai.
$(n + 1) - 1 = n$ woh exponent hai drop ke baad — exactly wahi power jo hum chahte the.

Toh hamara guess $x^{n}$ times ek unwanted factor $(n + 1)$ deta hai. Figure mein yeh ek target ( $x^{n}$ , plum) aur hamara arrow usse orange stretch-factor $(n + 1)$ se overshoot karta hua dikhta hai.

Step 4 — Divide karke overshoot fix karna

KYA HAI. Guess ko us constant se divide karo: $\frac{d}{d x} [\frac{x ^{n + 1}}{n + 1}] = \frac{( n + 1 ) x ^{n}}{n + 1} = x^{n} . ✓$

Front par $\frac{1}{n + 1}$ ek constant hai — differentiation constants ko untouched carry karta hai.
Differentiation se produce hua $(n + 1)$ , humne jo $\frac{1}{n + 1}$ rakha tha usse milta hai aur dono $1$ mein cancel ho jaate hain.
Jo bachta hai woh ek clean $x^{n}$ hai — exactly wahi slope jo humne order ki thi.

KYUN. Isliye power rule kehta hai "power ko raise karo, phir new power se divide karo." Har verb Step 2 ke differentiation ke do moves mein se ek ko undo karta hai.

PICTURE. Balance-scale figure: guess side par orange factor $(n + 1)$ ek plum $\frac{1}{n + 1}$ weight se cancel hota hai, aur scale $x^{n}$ par settle ho jaata hai.

Abhi tak humne dikhaya hai ki $\frac{x ^{n + 1}}{n + 1}$ ek aisi function hai jiska slope $x^{n}$ hai. Bilkul agle step mein hum dekhenge ki yeh wahi ek hi nahi hai — uski har vertical shift bhi kaam karti hai — aur wahi cheez neeche diye boxed rule mein extra " $+ C$ " ko force karti hai. Abhi ke liye rule ko ek placeholder ke saath padhein: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + (some constant, explained in Step 5), n \neq = - 1.$

Step 5 — $+ C$ kahan se aata hai? (lost-constant picture)

KYA HAI. $y = x^{2}$ , $y = x^{2} + 3$ , aur $y = x^{2} - 2$ dekho. Kisi bhi fixed $x$ par unke tangent rulers parallel hain — identical slope $2 x$ .

EXTRA CONSTANT KYUN ZAROORI HAI. Differentiation ek constant ko zero kar deta hai: $\frac{d}{d x} (x^{2} + 3) = 2 x, \frac{d}{d x} (x^{2} - 2) = 2 x .$ $+ 3$ aur $- 2$ slope mein koi trace nahi chhodh'te. Peeche chalte waqt, hum unhe recover nahi kar sakte, toh hum ek single letter $C$ likhte hain — constant of integration — jo represent karta hai "koi constant jo mujhe pata nahi." Yeh wahi " $C$ " hai jo har antiderivative mein aata hai.

PICTURE. Parabola ke teen orange copies, vertically shifted, same $x$ par teal tangent rulers ke saath — sab ek hi direction mein point karte hue. Plum bracket vertical gaps ko label karta hai " $C$ : neeche aate waqt kho gaya."

Ab boxed rule apna final symbol earn karta hai:

Step 6 — Forbidden case: $n = - 1$ rule ko kyun tod deta hai

KYA HAI. Power rule mein $n = - 1$ rakho: $\frac{x ^{n + 1}}{n + 1} n = - 1 \frac{x ^{0}}{0} = \frac{1}{0} = undefined.$

$n + 1$ , $- 1 + 1 = 0$ ban jaata hai.
$0$ se divide karna meaningless hai — poora formula collapse ho jaata hai.

GEOMETRICALLY KYUN FAIL HOTA HAI. $n = - 1$ ke liye integrand $\frac{1}{x}$ hai. Power raise karne se $x^{0} = 1$ milta hai, jiska slope $0$ hai, $\frac{1}{x}$ nahi. "Raise-then-divide" trick literally $\frac{1}{x}$ slope produce kar hi nahi sakti — iske liye ek alag function chahiye.

Patch. Woh function jiska slope $\frac{1}{x}$ hai, natural logarithm hai (dekho Natural log and exponential functions). Kyunki $\frac{1}{x}$ negative $x$ ke liye bhi exist karta hai jabki $ln x$ nahi karta, hum $ln ∣ x ∣$ use karte hain: $\int \frac{1}{x} d x = ln ∣ x ∣ + C .$

PICTURE. Left panel: powers ki smooth staircase mein $n = - 1$ par ek hole punched. Right panel: plum curve $y = ln ∣ x ∣$ woh hole plug karta hua — uske tangent slope $x$ badhne par sikte hue dikhaye gaye, $\frac{1}{x}$ ko match karte hue.

Step 7 — Chhupa hua fine print: $x$ ki kaunsi values allowed bhi hain?

KYA HAI — teen tarah ke exponents.

$n$ ek whole number $\geq 0$ (jaise $x^{2}, x^{5}$ ): $x^{n}$ har real $x$ ke liye defined hai, negatives aur zero bhi. Rule sab $x$ ke liye hold karta hai.
$n$ ek negative whole number $\leq - 2$ (jaise $x^{- 3}$ ): sab $x \neq = 0$ ke liye defined — zero se divide nahi kar sakte. Rule zero ko chhodh ke har jagah hold karta hai.
$n$ ek fraction / non-integer (jaise $x^{1/2} = x$ ): yahan subtle part hai. $x$ poochhta hai "kaun sa number square ho ke $x$ deta hai?" — $x < 0$ ke liye real numbers mein yeh possible nahi. Toh $x^{1/2}$ (aur koi bhi $x^{n}$ jahan $n$ non-integer ho) sirf $x \geq 0$ ke liye defined hai.

INTEGRAL KE LIYE YEH KYUN MATTER KARTA HAI. Identity $\frac{d}{d x} \frac{x ^{n + 1}}{n + 1} = x^{n}$ sirf wahin check ki ja sakti hai jahan dono sides exist karti hain. Non-integer $n$ ke liye woh region $x \geq 0$ hai (aur aksar $x > 0$ , kyunki $x = 0$ par ek negative fractional power blow up kar deta hai). Toh honest statement hai: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C valid on x > 0 when n is not a whole number.$

$x < 0$ tak extend karna. Odd roots ke liye negatives ka ek meaning hota hai — jaise $3 - 8 = - 2$ , toh $x^{1/3}$ sab real $x$ ke liye defined hai, aur rule wahin extend hota hai. Lekin even roots ( $x$ , $x^{1/4}$ , …) ke liye $x < 0$ mein koi real value exist nahi karti, toh antiderivative ka wahin simply koi real meaning nahi hota — aapko complex numbers chahiye hoge, jo ek alag course hai. Safe habit: jab bhi aap kisi root ko fractional power mein rewrite karo, turant allowed $x$ note karo.

PICTURE. Teen number lines stacked. Top (whole $n \geq 0$ ): poori line teal "allowed" hai. Middle (negative whole $n$ ): $x = 0$ par plum hole ke saath teal har jagah. Bottom (even root wala fractional $n$ ): sirf $x \geq 0$ teal hai; $x < 0$ greyed "no real value."

Step 8 — Do quick worked checks

Yahan ke tools composite aur product integrands tak Integration by substitution aur Integration by parts ke zariye extend hote hain; limits add karne se yeh antiderivatives Definite integrals & FTC ke zariye areas ban jaati hain.

Ek-picture summary

Poori derivation ko ek single loop ki tarah padho: differentiate power giraata hai aur multiply karta hai (orange arrow); integrate power uthata hai aur divide karta hai (plum arrow); vertical shift $+ C$ woh cheez hai jo loop recover nahi kar sakta; point $n = - 1$ woh akela jagah hai jahan loop toot jaata hai ( $ln ∣ x ∣$ se patch hota hai); aur fractional $n$ quietly allowed $x$ restrict kar deta hai.

Recall Feynman retelling — kisi dost ko batao

Kisi power ko differentiate karna do chote kaam karta hai: exponent ko front par multiplier ki tarah slide karta hai, aur exponent ko ek se knock down karta hai. Integrate karna sirf ek detective ka kaam hai jo un do cheezon ko reverse mein undo karta hai: exponent ko ek se upar uthao, phir us naye exponent se divide karo taaki jo multiplier aata woh khatam ho jaaye. Kyunki graph ko upar ya neeche karna kabhi uski steepness nahi badalta, detective kabhi nahi bata sakta ki aap kis vertical shift se shuru hue — woh anjaan shift hi " $+ C$ " hai. Woh akeli jagah jahan trick khud ko destroy karti hai woh $n = - 1$ hai: $- 1$ ko $0$ tak raise karna aur $0$ se divide karna nonsense hai, auraise bhi $x^{0} = 1$ ki slope flat hoti hai, $\frac{1}{x}$ nahi — toh nature woh akela gap $ln ∣ x ∣$ se bharta hai. Ek aakhri fine print: agar power ek fraction hai jaise $\frac{1}{2}$ , toh $x^{n}$ sirf $x \geq 0$ ke liye sense karta hai (negative ka square root nahi nikal sakte), toh answer sirf us side zero pe hi rehta hai.

Connections

4.2.02 Basic integration rules — power, trig, exponential, log (Hinglish) — parent topic
Derivatives — basic rules — woh forward machine jise humne reverse kiya
Integration by substitution · Integration by parts — composites aur products ke liye
Definite integrals & FTC — in antiderivatives ko areas mein badalta hai
Natural log and exponential functions — $ln ∣ x ∣$ patch ka source
Trigonometric integrals — trig ke liye wahi "ulta padho" wala idea

Visual walkthrough — Basic integration rules — power, trig, exponential, log