4.2.2 · D1Basic integration rules — power, trig, exponential, log
Integration kuch aur nahi balki differentiation ko ulta chalana hai: tumhe ek rate of change diya jata hai aur puchha jata hai ki woh kis function se aayi. Us machine ko backwards padhne ke liye pehle tumhe forward machine — notation, pictures, aur rules — poori tarah samajhni hogi — isliye yeh page woh har symbol build karta hai jo parent note quietly assume karta hai, ek blank page se shuru karke.
Isse pehle ki tum "derivative rules ko backwards padho" enjoy karo, tumhe yakeen hona chahiye ki page par har squiggle ka matlab kya hai. Neeche, har symbol unpack kiya gaya hai: seedhe words → picture → aur topic ko uski zaroorat kyun hai. Upar se neeche padho; koi cheez use nahi hoti jab tak build nahi ho jaati.
Ek function ek rule hai jo ek input number leta hai aur exactly ek output number deta hai. Hum f ( x ) f x f x
Iska picture ek graph hai: har horizontal position x f ( x ) hota hai — ek height jo is baat par depend karti hai ki tum kahan khade ho.
Intuition Topic ko iska kya zaroorat haiHar integral ∫ f ( x ) d x f f ( x ) f ( x )
Definition Variable of integrationx input hai jise hum change karne ke liye free hain . Integral ke end mein chhota d x "woh variable jiske upar main integrate kar raha hoon woh x — yeh sirf knob ka naam batata hai (abhi ke liye).
Socho ek slider jise tum horizontal axis ke saath left aur right push kar sakte ho. Woh slider hi x d x
Intuition Topic ko iska kya zaroorat hai∫ … d x ∫ … d t kaun sa letter moving hai aur kaun se letters frozen constants hain. Agar yeh miss kiya toh tum variable aur constant mein farak nahi kar sakte.
Ek seedhi line ka slope yeh hai ki "jab tum ek step right lete ho toh height kitni badhti hai": slope = run rise 0
Ek curve ke liye, slope chalte waqt badalta rehta hai. Har point par hum tangent line kheenchte hain — woh seedhi line jo curve ko wahan just kiss karti hai — aur uska slope padhte hain.
f derivative , likha d x d f ( x ) f ′ ( x ) nayi function hai jiska value har x f x d x d
Intuition Topic ko iska kya zaroorat hai — yeh POORA topic ulta haiParent note ka motto hai "integration is anti-differentiation." Woh sentence meaningless hai jab tak d x d f ′ ( x ) har jagah ki steepness di gayi ho, original curve wapas banao.
Forward rules ke liye Derivatives — basic rules dekho; har integral rule unme se ek ulti hai.
f ′ ( x ) f x exactly wahi hai jo d x d f ( x ) f d x d
Parent note dono likhta hai — jaise "F ′ ( x ) = f ( x ) d x d sin x = cos x d x d kaun sa variable hai is baat ke baare mein loud rehna chahte ho; f ′
∫ f ( x ) d x f x ∫ stretched letter S hai (yeh kabhi Sum ke liye tha). Abhi ke liye, poori cheez ko ek sawaal ki tarah treat karo: "kaun si function ka slope f ( x )
Definition Constant of integrationAnswer F ( x ) + C + C family of parallel curves , sab ki slope har x
+ C shape batata hai, kabhi height nahi. Figure mein teen curves hain x 2 x 2 + 7 x 2 − 4 x 2 x 0 + C
Ek antiderivative F F ′ ( x ) = f ( x ) + C
Definition Powers, roots, aur negative/fraction exponents
x n x n n x − 1 = x 1 x − k = x k 1 negative exponent matlab reciprocal (palat do).x 1/2 = x x 1/2 x fraction exponent matlab root .
Intuition Topic ko iska kya zaroorat haiPower rule ∫ x n d x = n + 1 x n + 1 + C kuch bhi x kuch x → x 1/2 x 3 4 → 4 x − 3
Worked example Rewrite drills (parent ke examples se)
x = x 1/2 x 3 4 = 4 x − 3 x 6 = 6 x − 1 n = − 1
Definition Absolute value∣ x ∣ x ∣5∣ = 5 ∣ − 5∣ = 5 distance samjho, direction ignore karte hue.
Log rule hai ∫ x 1 d x = ln ∣ x ∣ + C x 1 x ln x ∣ x ∣ sab x = 0
e ln e ≈ 2.718 ln x e x e x undo karta hai. Iska slope famous taur par d x d ln x = x 1
Intuition Topic ko iska kya zaroorat haiDo parent rules is par depend karte hain: ∫ e x d x = e x + C e x ∫ a x d x = ln a a x + C Natural log and exponential functions mein theek se milo.
Definition Sine aur cosineEk point ko radius 1 cos x horizontal position hai, sin x vertical position, jahan x − 1 1 doosre mein badal jaate hain.
Definition Family ke baaki members
tan x = cos x sin x sec x = cos x 1 csc x = sin x 1 cot x = sin x cos x
Intuition Topic ko iska kya zaroorat hai — aur minus-sign trapForward slopes hain d x d sin x = cos x d x d cos x = − sin x cosine side par minus ek hi fact hai jo parent ke "sign trap" ke peeche hai: ∫ sin x d x = − cos x + C + cos x cos x shrink karta hai, aur woh shrinking negative slope hai. Iske aur examples Trigonometric integrals mein hain.
"Linear" matlab do freedoms: tum ek sum ko alag-alag pieces mein split kar sakte ho, aur ek constant multiplier ko bahar nikal sakte ho. Kyunki differentiation d x d [ a f + b g ] = a f ′ + b g ′
Common mistake Tempting non-ruleIntegrals ke liye koi product rule ya quotient rule nahi hai. Aise lagta hai jaise differentiation ek de deta, lekin nahi deta — products ke liye Integration by parts aur composites ke liye Integration by substitution chahiye.
Function f of x as a height
Derivative slope function
Exponents powers roots negatives
Sine cosine on the circle
Linearity split and scale
Integration anti differentiation plus C
Har arrow same destination par jaata hai: parent topic bas yahi foundations hain, ulte padhe, aur + C
f ( x ) Horizontal axis par point x
Ek sentence mein derivative f ′ ( x ) Ek nayi function jo har x f
Kya f ′ ( x ) d x d f ( x ) Haan — identical meaning, slope-function ke liye do notations.
Kyun har indefinite integral + C Slope sirf shape fix karta hai, vertical height nahi; differentiation kisi bhi constant ko 0 kar deta hai, isliye hum use recover nahi kar sakte.
x x 3 4 x 6 x 1/2 4 x − 3 6 x − 1
Negative exponent x − k Reciprocal, x k 1
Log integral mein x ∣ x ∣ x 1 x ∣ x ∣ x = 0
ln x "e x e x
Unit circle par kaun sa coordinate cos x Moving point ka horizontal coordinate.
Kaun si trig slope minus sign carry karti hai? d x d cos x = − sin x ∫ sin x d x = − cos x
Linearity tumhe kaun si do freedoms deti hai? Sum ko alag-alag integrals mein split karo; constant multipliers ko bahar nikalo.
Kya integration ke liye koi product rule hai? Nahi — iske liye integration by parts ya substitution use karo.