Basic integration rules — power, trig, exponential, log

4.2.2 · Maths › Calculus II — Integration

Integration mein "+C" ki zaroorat kyun hoti hai?

Power Rule — scratch se nikala gaya

Derivation. Andaza lagao ki antiderivative $x^{n + 1}$ jaisa dikhta hai. Test karo: $\frac{d}{d x} x^{n + 1} = (n + 1) x^{n} .$ Ye $(n + 1)$ times zyada bada hai. Ye step kyun? Differentiation ne purane power+1 se multiply kar diya, toh hum ise $(n + 1)$ se divide karke cancel karte hain: $\frac{d}{d x} [\frac{x ^{n + 1}}{n + 1}] = \frac{( n + 1 ) x ^{n}}{n + 1} = x^{n} . ✓$

Trig Rules — derivatives ko ulta padho

$\int cos x d x$ ki derivation. Hum chahte hain $F$ jisme $F^{'} = cos x$ ho. Kyunki $\frac{d}{d x} sin x = cos x$ , $F = sin x$ lo. Ho gaya.

$\int sin x d x$ ki derivation. Hum chahte hain $F^{'} = sin x$ . Hum jaante hain $\frac{d}{d x} cos x = - sin x$ , jo hamari zaroorat ka negative hai. Ye step kyun? $- 1$ se multiply karo: $\frac{d}{d x} (- cos x) = sin x .$ Toh $\int sin x d x = - cos x + C$ .

Exponential Rule

$\int e^{x} d x = e^{x} + C .$

General base $a > 0$ ke liye: hum jaante hain $\frac{d}{d x} a^{x} = a^{x} ln a$ . Ye $ln a$ times zyada bada hai, toh $ln a$ se divide karo: $\int a^{x} d x = \frac{a ^{x}}{l n a} + C .$

Log Rule — $n = - 1$ ke liye patch

$\int \frac{1}{x} d x = ln ∣ x ∣ + C .$

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

Linearity — kyun hum split aur scale kar sakte hain

Worked Examples

Recall Feynman: ek 12 saal ke bacche ko samjhao

Differentiate karna aisa hai jaise poochho "ye kitni tezi se badal raha hai?" Integrate karna detective wali baat hai: main tumhe har pal ki speed batata hoon aur tum reconstruct karte ho ki gaadi kahan thi. Lekin ek dikkat hai — tumhe gaadi ka starting point nahi pata, toh har answer "+ koi anjaan starting point" ke saath khatam hota hai (ye hi $+ C$ hai). Integrate karne ke liye bas poochho "mujhe kya differentiate karna padta ye banane ke liye?" aur likh do — phir divide karo ya sign flip karo taaki bilkul sahi aaye.

Flashcards

+ C

kya represent karta hai aur kyun aana ZAROORI hai?

Kho gaya constant — differentiation kisi bhi constant ko 0 kar deta hai, toh antiderivative use recover nahi kar sakta;

+ C

saari possibilities ko encode karta hai.

Power rule aur uska ek exception batao.

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

jab

n \neq = - 1

;

n = - 1

par zero se divide ho jaata.

\int \frac{1}{x} d x = ?

aur

∣ x ∣

kyun?

ln ∣ x ∣ + C

; absolute value

x < 0

ke liye bhi validity badhata hai kyunki

\frac{d}{d x} ln ∣ x ∣ = \frac{1}{x}

har jagah 0 ke alawa.

\int sin x d x = ?

- cos x + C

(minus sign dhyan se).

\int cos x d x = ?

sin x + C

\int sec^{2} x d x = ?

tan x + C

\int e^{x} d x = ?

aur kyun unchanged?

e^{x} + C

;

e^{x}

differentiation ka fixed point hai.

\int a^{x} d x = ?

\frac{a ^{x}}{l n a} + C

\frac{d}{d x} a^{x}

se extra

ln a

divide karke nikala.

Kya integration mein product rule hota hai?

Nahi — products ke liye substitution ya integration by parts chahiye.

Integral verify kaise karte hain hamesha?

Apna answer differentiate karo; woh integrand wapas dena chahiye.

Connections

Derivatives — basic rules (yahan har rule inhi mein se ek ulta hai)
Integration by substitution (composite functions handle karta hai)
Integration by parts (products handle karta hai)
Definite integrals & FTC (ye antiderivatives + limits use karta hai)
Natural log and exponential functions ( $ln ∣ x ∣$ aur $e^{x}$ rules ka source)
Trigonometric integrals (basic trig table ko extend karta hai)