Grade 11-12 (age 16-18)

high school Indefinite Integral

An indefinite integral runs differentiation backwards: it looks for the original function whose derivative is the given one. A constant becomes 0 when differentiated and vanishes, so there is not one such function but endlessly many, each differing by a constant. That is why the result of an indefinite integral always carries a constant of integration C to fill the place of the lost constant. In the first picture, changing C moves the curve x² + C up or down without changing its shape, and the tangent slope at x = 1 is 2 on every curve.

Integration undoes differentiation

🔄 The idea of the indefinite integral

①Find a function F(x) whose derivative is f(x)

②Such an F(x), with F'(x) = f(x), is an indefinite integral of f(x)

③Adding any constant to F(x) leaves the derivative unchanged, so there are endlessly many answers

An indefinite integral is a family of curves

Constant of integration C0

Definition of the indefinite integral

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

It stands for all antiderivatives of f(x); C is the constant of integration

💡 What the constant C means

①C is fixed only when a condition such as a starting value is given

②For example, integrating the velocity v(t) = 2t gives the position s(t) = t² + C, with C = s(0)

③When finding an indefinite integral, never leave out + C

Differentiation and integration undo each other

Exponent n2

Power Rule for Integration

∫ xⁿ dx = xⁿ⁺¹n+1 + C (n ≠ -1)

The power rule (xⁿ)' = nxⁿ⁻¹ used backwards

Basic Integration Rules

Constant multiples, sums and differences

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

Constant multiples come out of the integral; sums and differences are integrated term by term

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 (n ≠ -1)

Summary

Indefinite Core

∫ xⁿ dx = xⁿ⁺¹n+1 + C (n ≠ -1)

Undo differentiation and add the constant C

🎯 Exam Points

①An indefinite integral undoes differentiation; always add + C

②Integrating xⁿ raises the exponent to n + 1 and puts 1/(n + 1) in front

③Integrate a polynomial term by term and add

④A condition such as a starting value fixes C

⑤Check: differentiating your answer should give back the original function

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 constant of integration C at the end.

∫ (3x² + 2x) dx = x³ + x² + C

▸ x³ + x² + C

Differentiating x³ + x² + C gives back 3x² + 2x, which confirms the answer.

Example 2

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

Integrating 2x + 1 gives F(x) = x² + x + C.

F(x) = x² + x + C

Use F(0) = 3 to fix C.

F(0) = C = 3 ⟹ F(x) = x² + x + 3

▸ F(x) = x² + x + 3

This picks out, from the whole family of curves, the one that passes through (0, 3). One condition fixes one 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

Integrating term by term, 6x² becomes 2x³ and 4x becomes 2x².

∫ 6x² dx = 2x³, ∫ 4x dx = 2x²

Keep the minus sign and add the constant C.

2x³ - 2x² + C

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Applications of Derivatives

Definite Integral

Last updated 2026-10-17

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