Indefinite Integral

The indefinite integral asks what function you get when you undo differentiation. You want the original function whose derivative is the given one, but any constant vanishes when differentiated, so there are infinitely many answers. That is why the result always carries a constant of integration C to stand in for that lost constant. Drag the constant C slider and watch the x² + C curves slide up and down together, keeping the same shape.

Integration as Reverse of Differentiation

🔄 Core Idea

①"Which function, when differentiated, gives the original?"

②Inverse of differentiation: find F(x) with F'(x) = f(x)

③Adding any constant C disappears under differentiation → infinitely many answers!

Antiderivative Family

Integration constant C0

Definition

∫ f(x) dx = F(x) + C (where F'(x) = f(x))

All antiderivatives of f(x); C is the integration constant

💡 Meaning of C

①C is fixed by the initial condition

②e.g., integrate v(t) = 2t → position s(t) = t² + C with C = s(0)

③Forgetting C costs marks!

Inverse Relationship of Diff & Integral

Power n2

Power Rule for Integration

∫ xⁿ dx = xⁿ⁺¹n+1 + C (n ≠ -1)

Inverse of (xⁿ)' = nx^n-1

Basic Integration Rules

Constant Multiple, Sum, Difference

∫ kf(x) dx = k∫ f(x) dx, ∫ [f ± g] dx = ∫ f dx ± ∫ g dx

Constants pull out; sums and differences split

Constant

∫ k dx = kx + C

Linear

∫ x dx = x²/2 + C

Quadratic

∫ x² dx = x³/3 + C

Power n

∫ xⁿ dx = xⁿ⁺¹n+1 + C

Wrap-up

Indefinite Core

∫ xⁿ dx = xⁿ⁺¹n+1 + C

Inverse of differentiation + constant C

🎯 Exam Points

①Indefinite integral = inverse of differentiation; always add + C

②Power rule needs n ≠ -1

③Integrate polynomials term-by-term

④Initial conditions fix C (e.g., f(0) = 3)

⑤Verify: differentiating the result must give the integrand back

Worked Examples & Past Exam

Example 1

Find ∫ (3x² + 2x) dx.

Apply ∫ xⁿ dx = xⁿ⁺¹/(n+1) to each term.

∫ 3x² dx = x³, ∫ 2x dx = x²

Add the integration constant C.

∫ (3x² + 2x) dx = x³ + x² + C

▸ x³ + x² + C

Indefinite integration is the inverse of differentiation; always add the constant + C.

Example 2

If F'(x) = 2x + 1 and F(0) = 3, find F(x).

F(x) = ∫ (2x + 1) dx = x² + x + C.

F(x) = x² + x + C

Use F(0) = 3 to find C.

F(0) = C = 3 ⟹ F(x) = x² + x + 3

▸ F(x) = x² + x + 3

An initial condition pins down the integration constant C.

2023 CSAT Math type, adapted

What is ∫ (6x² − 4x) dx? (C is the integration constant)

①2x³ − 2x² + C

②2x³ − 4x² + C

③6x³ − 4x² + C

④x³ − x² + C

⑤12x − 4 + C

▸ ① 2x³ − 2x² + C

Integrate each term: ∫ 6x² dx = 2x³, ∫ 4x dx = 2x².

∫ 6x² dx = 2x³, ∫ 4x dx = 2x²

Attach the sign and the integration constant.

2x³ - 2x² + C

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Applications of Derivatives

Definite Integral

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