Antiderivative — definition, family of solutions (+C)

1. Definition

WHAT each piece means

$\int \cdots \, dx$ = "find every function whose derivative is the thing inside."
$F(x)$ = one particular antiderivative.
$+C$ = packages up the infinitely many others.

2. WHY there is a $+C$ — derived from scratch

We don't assume the $+C$ ; we prove it must be there and that it's the only freedom.

Step 1 — At least two antiderivatives differ by a constant always work. Suppose $F'(x)=f(x)$ . Let $G(x)=F(x)+C$ for any constant $C$ . $G'(x) = F'(x) + \frac{d}{dx}(C) = f(x) + 0 = f(x).$ Why this step? The derivative of a constant is $0$ , so adding any constant leaves the slope untouched — every $F+C$ is also an antiderivative.

Step 2 — There are NO others (the family is complete). Suppose $F$ and $G$ are both antiderivatives of $f$ . Define $H(x)=G(x)-F(x)$ . Then $H'(x) = G'(x) - F'(x) = f(x) - f(x) = 0.$ Why this step? A function with zero derivative everywhere on an interval has zero slope everywhere → it cannot rise or fall → it is constant (this is a consequence of the Mean Value Theorem). $\Rightarrow H(x) = C \quad\Rightarrow\quad G(x) = F(x) + C.$

Figure — Antiderivative — definition, family of solutions (+C)

3. Worked examples

4. Common mistakes (Steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine you only know how steep a hill is at every step, but not how high you started. You can draw the shape of the hill perfectly — but you could have started 1 metre up, or 100 metres up, and the shape would look identical. That unknown starting height is the $+C$ . To pin it down, someone has to tell you the height at just one spot.

5. Active-recall flashcards

What is an antiderivative of

f

A function

F

with

F'(x)=f(x)

on an interval.

Why does the indefinite integral carry

+C

Because

\frac{d}{dx}(C)=0

, any two antiderivatives differ only by a constant, so the whole family is

F(x)+C

Prove that two antiderivatives of the same

f

differ by a constant.

Let

H=G-F

; then

H'=G'-F'=f-f=0

, and zero derivative on an interval forces

H=

const (MVT).

\int x^2\,dx = ?

\dfrac{x^3}{3}+C

\int \cos x\,dx = ?

\sin x + C

For which

n

does the power rule

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

fail, and what replaces it?

n=-1

; then

\int x^{-1}dx=\ln|x|+C

When does

+C

safely cancel?

In a definite integral

\int_a^b

, since

[F+C]_a^b=F(b)-F(a)

Why

\ln|x|

(with absolute value) for

\int 1/x\,dx

Because

\frac{d}{dx}\ln|x|=1/x

for

x\neq0

, covering both signs.

What extra information picks one member of the family?

An initial/boundary condition like

F(x_0)=y_0

Geometrically, how are members of the family related?

Vertical translates with identical slope

f(x)

at every

x

("parallel" curves).

6. Connections

Mean Value Theorem — justifies "zero derivative ⇒ constant," the heart of the $+C$ proof.
Fundamental Theorem of Calculus — links antiderivatives to definite integrals.
Power Rule (Integration) — the $n=-1$ exception.
Differential Equations — Initial Value Problems — using a condition to fix $C$ .
Logarithmic & Exponential Integrals — origin of $\ln|x|$ .
Definite Integral — where $+C$ harmlessly cancels.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, antiderivative ka matlab simple hai: differentiation ulta karna. Agar tumhe slope $f(x)$ pata hai har point pe, toh original function $F$ dhoondhna — jiska $F'(x)=f(x)$ ho — yahi antiderivative hai. Lekin ek twist hai: sirf slope se tum height nahi bata sakte. Curve ko upar-neeche shift karne se uska slope kahin change nahi hota. Isiliye answer ek nahi, balki ek poora family hota hai, aur wahi extra freedom hum $+C$ likhke capture karte hain.

$+C$ kyun aata hai, iska proof bhi seedha hai. Constant ka derivative $0$ hota hai, toh $F+C$ bhi antiderivative banega. Aur agar do antiderivatives $F$ aur $G$ ho, toh unka difference $H=G-F$ ka derivative $0$ aata hai — aur ek interval pe zero slope ka matlab function constant hai (yeh Mean Value Theorem se aata hai). Toh $G=F+C$ . Matlab family complete hai, isse zyada koi aur antiderivative nahi.

Practical baat: indefinite integral mein $+C$ kabhi mat bhoolna — warna tum keh rahe ho answer unique hai jabki infinite curves hain. Sirf definite integral $\int_a^b$ mein $C$ cancel ho jaata hai. Aur ek condition de di jaaye, jaise $F(0)=5$ , toh us se $C$ ki exact value nikal jaati hai — yani family mein se ek hi curve select ho jaata hai. Yeh idea aage differential equations aur Fundamental Theorem of Calculus mein bohot kaam aayega.