Antiderivative — definition, family of solutions (+C): Question bank

4.2.1 · D5 · Maths › Calculus II — Integration › Antiderivative — definition, family of solutions (+C)

Shuru karne se pehle, teen load-bearing facts yaad karo jis par yahan sab kuch tika hai:

$\frac{d}{d x} (C) = 0$ — ek constant ka slope zero hota hai, isliye use add karne se shape ke baare mein koi information nahi chhupti.
Ek interval par zero derivative ⇒ constant — yeh Mean Value Theorem hai jo heavy lifting kar raha hai.
$+ C$ hi ek maatra freedom hai; do antiderivatives ek connected interval par sirf isi se alag hote hain, aur kisi cheez se nahi.

True or false — justify

True or false: Agar

F^{'} (x) = G^{'} (x)

ek interval mein sab

x

ke liye hai, toh

F (x) = G (x)

hai.

False. Equal slopes force karte hain ki

F

aur

G

ek constant se alag hon, identical nahi —

F - G

ka zero derivative hai, toh yeh koi aisa constant hai jo nonzero bhi ho sakta hai.

True or false:

\int f (x) d x

ek single function ko name karta hai.

False. Yeh ek puri family

F (x) + C

ko name karta hai;

+ C

infinitely many parallel curves ko package karta hai, toh koi ek curve "the" answer nahi hai.

True or false: Kyunki

C

differentiate karne par vanish ho jaata hai, ise likhna optional bookkeeping hai.

False.

C

differentiate karne ke baad invisible hota hai lekin yeh ek indefinite integral ka poora content hai — ise drop karna ek unique answer claim karna hai jahan infinitely many exist karte hain.

True or false: Har function ka ek antiderivative hota hai jo tum elementary functions se likh sako.

False. Antiderivative hona (continuous

f

ke liye Fundamental Theorem of Calculus se guaranteed) alag baat hai aur use elementary form mein express kar paana alag baat — e.g.

e^{- x^{2}}

ka antiderivative hai lekin koi elementary formula nahi hai.

True or false: Agar

f

ke do antiderivatives ek point par agree karte hain, toh woh us interval par har jagah agree karte hain.

True. Woh ek constant

C

se differ karte hain; agar woh ek point par agree karte hain toh waahan

C = 0

hai, aur ek constant har jagah same hota hai, isliye woh poore interval par coincide karte hain.

True or false:

F (x) + C

ke graphs alag-alag

C

ke liye ek doosre ko cross kar sakte hain.

False. Yeh pure vertical translates hain; ek fixed vertical shift kabhi do parallel curves ko milne nahi deta. Cross hone ke liye unhe kisi

x

par ek hi height share karni padti, jo constant gap ko contradict karta.

True or false:

\int 0 d x = 0

False. Zero function ke antiderivatives exactly constants hain, isliye

\int 0 d x = C

— shape flat hai, lekin height abhi bhi unknown hai.

True or false: Ek definite integral

\int_{a}^{b} f d x

par,

C

ki choice answer ko change karti hai.

False.

[F + C]_{a}^{b} = (F (b) + C) - (F (a) + C) = F (b) - F (a)

; constant cancel ho jaata hai, yahi exactly wajah hai ki definite integrals single numbers hote hain (dekho Definite Integral).

Spot the error

Flaw dhundho: "

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

har real

n

ke liye."

Yeh

== n = - 1 ==

par toot jaata hai, jahan

n + 1 = 0

division by zero deta hai. Woh gap exactly wahin hai jahan logarithm aata hai:

\int x^{- 1} d x = ln ∣ x ∣ + C

(dekho Logarithmic & Exponential Integrals).

Flaw dhundho: "

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

Domain

x \neq = 0

mein

x < 0

ln x

ke under exclude kiya gaya hai. Dono signs par valid correct antiderivative

ln ∣ x ∣ + C

hai, kyunki

\frac{d}{d x} ln ∣ x ∣ = 1/ x

sab

x \neq = 0

ke liye.

Flaw dhundho: "

\frac{d}{d x} ln ∣ x ∣ = 1/ x

, aur

1/ x

kahin bhi risky nahi define hota, toh ek constant

C

poore answer ko cover karta hai."

Domain do disconnected intervals mein split ho jaata hai

(- \infty, 0)

aur

(0, \infty)

. "Only-a-constant" theorem ko ek single connected interval chahiye, isliye har piece ko apna constant

C_{1}, C_{2}

milta hai.

Flaw dhundho: "

H^{'} (x) = 0

domain

{x \neq = 0}

par har jagah hai, isliye

H

ek single constant hai."

Zero derivative constancy force karta hai sirf ek connected interval par.

x = 0

par hole ke across do pieces alag heights par baith sakti hain, isliye

H

left par ek constant aur right par doosra constant ho sakta hai.

Flaw dhundho: "

\int cos x d x = sin x

, done."

+ C

missing hai: correct family

sin x + C

hai. Kisi condition jaise

F (0) = 5

ke bina tum ek curve pin nahi kar sakte, isliye

C

chhod dena ek incomplete (aur technically galat) answer hai.

Flaw dhundho: "Kyunki differentiation ek function hai, integration uska inverse function hona chahiye."

Differentiation one-to-one nahi hai — yeh ek poori family

F + C

ko same

f

par map karta hai — isliye iska koi true inverse function nahi hai. "Antiderivative" family recover karta hai, single input nahi.

Edge cases

Edge case:

f (x) = 0

ka general antiderivative kya hai?

Koi bhi constant:

\int 0 d x = C

. Curve bilkul flat hai, lekin uski height abhi bhi undetermined hai, isliye family sab horizontal lines hain.

Edge case: Domain

x \neq = 0

par,

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

mein kitne free constants hain sach mein?

Do, ek har connected interval ke liye:

ln (- x) + C_{1}

x < 0

ke liye aur

ln x + C_{2}

x > 0

ke liye. Ek textbook ka "

ln ∣ x ∣ + C

" quietly assume karta hai ki tum ek hi side par ho.

Edge case: Agar

f

sirf isolated points par define hai (kisi interval par nahi), toh kya "antiderivative" ka koi matlab bhi hai?

Usual sense mein nahi — definition require karti hai ki

F^{'} = f

ek interval par ho, kyunki slope ek limiting notion hai jisko nearby points chahiye. Isolated points differentiate karne ke liye koi interval nahi dete.

Edge case: Kya

f

ke do antiderivatives ek hi interval par aise graphs ho sakte hain jo ek point par tangent hon (bina cross kiye touch karein)?

Nahi — woh ek constant se differ karte hain, isliye unka vertical gap fixed hai. Touch hone ke liye gap ko ek point par zero tak pahunchna hoga jabki kahin aur nonzero rahna hoga, jo ek constant nahi kar sakta.

Edge case: Ek continuous

f

ke liye

[a, b]

par, kya antiderivative ka exist karna guaranteed hai chahe tum formula na likh sako?

Haan. Fundamental Theorem of Calculus ek

G (x) = \int_{a}^{x} f (t) d t

ki tarah banata hai; existence continuity se guaranteed hai, is baat se independent ki closed-form expression available hai ya nahi.

Recall Yahan har trap ki one-line summary

Is page par har trap teen ideas mein se ek hai disguise mein: (1) slope height hide karta hai → hamesha $+ C$ ; (2) zero slope matlab constant sirf EK connected interval par → domain holes ke liye dekho; (3) power rule $n = - 1$ par mar jaata hai → logarithm aata hai.

Connections

Mean Value Theorem — "zero derivative ⇒ constant" ke peeche ka engine.
Power Rule (Integration) — $n = - 1$ trap.
Logarithmic & Exponential Integrals — $ln ∣ x ∣$ kahaan se aata hai aur kyun absolute value.
Differential Equations — Initial Value Problems — kaise ek condition $C$ fix karti hai.
Fundamental Theorem of Calculus — antiderivatives ka existence, definite integrals se link.
Definite Integral — jahan $+ C$ harmlessly cancel ho jaata hai.

Question bank — Antiderivative — definition, family of solutions (+C)

True or false — justify

Spot the error

Why questions

Edge cases

Connections