Indefinite Integrals
FoundationAn indefinite integral reverses differentiation: recover the original function from its derivative and add the constant of integration C; master the power, reciprocal, exponential and trigonometric formulas plus linearity and substitution, and you can rebuild curves from rates — the road to definite integrals.
- Indefinite integration reverses differentiation: ∫f(x)dx is the family of all F with F'(x) = f(x).
- The constant C corresponds to the family of curves; use a condition to find C when given, otherwise keep C — write plus C the moment you integrate.
- Power rule: raise the exponent by one and divide by the new power; reciprocals give ln x; eˣ, sin, cos and sec² each have a matching formula.
文體小知識：數學概念說明文——由求導的逆過程引入不定積分與積分常數 C，整理冪、倒數、指數與三角的積分公式、線性性質與代換積分法，並應用於由變率還原曲線。
§Text Passage
1The last lesson went from a function to its derivative; this one walks backwards: given the derivative, recover the original function. The process is called indefinite integration, and the curriculum states explicitly that it is the reverse of differentiation. The notation is the elongated S sign followed by f(x) dx, where f(x) is the integrand and dx marks the variable of integration. Integration and differentiation are inverse operations, just like addition and subtraction, or powers and roots.
2Why 'indefinite'? Because adding a constant to any function leaves its derivative unchanged: x squared, x squared plus one, x squared minus seven all differentiate to two x. So recovering a function from its derivative gives not one curve but a family of vertically shifted curves, and an arbitrary constant C — the constant of integration — must be added. DSE examiners penalise a missing C every year; build the habit of writing plus C the moment you integrate.
3The basic formulas mirror the differentiation table. The power rule is the core: to integrate x to the power n, raise the power by one and divide by the new power, giving x to the (n+1) over (n+1), then add C. The only exception is the reciprocal: the integral of one over x is the natural logarithm ln x plus C. For example, integrating x to the fourth gives x to the fifth over five; rewrite the square root of x as x to the one half before integrating.
4Exponential and trigonometric functions have matching formulas: the integral of e to the x is itself; the integral of sine is minus cosine, of cosine is sine, and of secant squared is tangent — all plus C. The standard way to check an integral is to differentiate the result and see whether the integrand returns; this both verifies the answer and shows how integration and differentiation lock into each other.
5Linearity makes multiple terms easy: constants factor out of the integral, sums and differences split into separate integrals, and the whole expression needs only one C. When the integrand is composite, use substitution: let the inner function be u, convert dx into du, integrate the simpler form, and substitute back. The curriculum includes substitution but not integration by parts. In applications, recovering a curve from its tangent slope and one point is the most typical use of the indefinite integral.
1Summary 重點摘要
An indefinite integral reverses differentiation: recover the original function from its derivative and add the constant of integration C; master the power, reciprocal, exponential and trigonometric formulas plus linearity and substitution, and you can rebuild curves from rates — the road to definite integrals.
2Vocabulary cards Vocabulary
The family of all antiderivatives of f(x), written ∫f(x)dx, with the constant of integration C.
A function F(x) whose derivative is f(x), i.e. F'(x) = f(x).
An arbitrary constant C representing the family of antiderivative curves shifted vertically.
The function f(x) being integrated in ∫f(x)dx.
∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ −1): raise the power by one, then divide by the new power.
∫kf(x)dx = k∫f(x)dx and ∫[f(x)±g(x)]dx = ∫f(x)dx ± ∫g(x)dx.
Letting the inner function be u, converting dx into du, integrating the simpler form and substituting back; integration by parts is outside the curriculum.
The curriculum's characterisation of indefinite integration: it reverses differentiation, each undoing the other.
A given function value (such as a point on the curve) used to pin down the constant of integration C.
The logarithm to base e, written ln x; ∫(1/x)dx = ln x + C.
3Step by step Steps
- Step 1: Integration reverses differentiationRecovering a function from its derivative means constantly asking: which function differentiates to this? For example, x² has derivative 2x, since (x²)' = 2x; but (x²+1)' and (x²+5)' are also 2x, so the answer is not unique — we write ∫2x dx = x² + C. The notation ∫f(x)dx reads 'the integral of f(x) with respect to x'; ∫ is an elongated S, from the idea of summation, and dx marks the variable of integration. The curriculum requires this notation and an understanding that integration and differentiation are inverse processes.
- Step 2: The meaning of the constant CAdding a constant to any function leaves the derivative unchanged, so an indefinite integral gives a family of curves with identical shapes, shifted vertically. For example, ∫x² dx = x³/3 + C: taking C = 0, 1, −1 yields three vertically shifted parabolas. DSE examiners penalise a missing C every year: whenever the answer is an indefinite integral, plus C is compulsory; and if the question supplies a condition — a point on the curve, or a value at some time — then C can be determined uniquely, which is exactly how applied questions are set.
- Step 3: Integrating powers and reciprocalsThe core formula is ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1: raise the power by one, then divide by the new power. The reciprocal is the exception: ∫(1/x)dx = ln x + C. For example, ∫x⁴ dx = x⁵/5 + C; ∫√x dx = ∫x^½ dx = x^(32)÷(32) + C = (23)x^(32) + C; ∫x^(−2) dx = x^(−1)÷(−1) + C = −1/x + C. Converting surds and fractions to powers is the fixed move before applying the formula, and the n = −1 exception must be memorised separately.
- Step 4: Integrating exponentials and trigonometric functionsThese mirror the differentiation table: ∫eˣ dx = eˣ + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C. Note that the integral of sine carries a minus sign, because (−cos x)' = sin x. For example, integrating cos x gives sin x plus C; checking means differentiating sin x and recovering cos x. Differentiating your result to see whether the integrand returns is the standard self-check for integration — thirty seconds in an exam that can save the whole question.
- Step 5: LinearityConstants factor out and sums split apart: ∫kf(x)dx = k∫f(x)dx and ∫[f(x)±g(x)]dx = ∫f(x)dx ± ∫g(x)dx. For example, ∫(6x² − 4x + 5) dx = 6×(x³/3) − 4×(x²/2) + 5x + C = 2x³ − 2x² + 5x + C — a single C at the very end. Integrating a polynomial means applying the power rule term by term and merging the constants; watch the order of multiplying and dividing coefficients, e.g. six times x cubed over three gives two x cubed.
- Step 6: The substitution methodWhen the integrand is composite, let the inner function be u, convert du and dx into each other, integrate the simpler form, then substitute back. For example, ∫(2x+1)⁵ dx: let u = 2x+1, so du = 2dx, i.e. dx = du/2; the integral becomes ∫u⁵(du/2) = u⁶/12 + C = (2x+1)⁶/12 + C. The steps are fixed: set u, find du, substitute, integrate, reverse-substitute. The curriculum includes substitution but not integration by parts; whenever a bracket or root hides a function, first ask whether substitution turns it into a simple power.
- Step 7: Application: rebuilding a function from its rateApplied questions often give the tangent slope (the derivative) and a point, asking for the curve: integrate to get the family with C, then substitute the condition to determine C. For example, a curve has slope 6x² and passes through (1, 5): integrating gives y = 2x³ + C; substituting x = 1, y = 5 gives 2 + C = 5, so C = 3 and the curve is y = 2x³ + 3. Such questions chain integration, the constant and the condition into one line; write C when integrating, substitute, solve for C, and state the unique curve — none of these steps can be skipped.
5Worked example Example
Example: the tangent to a curve has slope 6x² − 4x + 1 at every point, and the curve passes through (1, 4). Find the equation of the curve.
- Let the curve be y = f(x); then f'(x) = 6x² − 4x + 1
- Integrate: y = ∫(6x² − 4x + 1) dx = 2x³ − 2x² + x + C
- Substitute (1, 4): 2(1)³ − 2(1)² + 1 + C = 4, i.e. 1 + C = 4, so C = 3
- Verify: y' = 6x² − 4x + 1, matching the given slope
- Answer: The curve is y = 2x³ − 2x² + x + 3. Full solution: state f'(x), integrate keeping C, substitute the given point to find C, and verify by differentiating; a missing C, or failing to use the condition to find C, loses marks.
6Common mistakes Mistakes
7Quick check Quick check
Q1Why must the answer to an indefinite integral include +C?
Q2Evaluate ∫(4x³ − 6x) dx.
Q3Evaluate ∫(eˣ + 1/x) dx for x > 0.
Q4Evaluate ∫(3x+2)⁴ dx.
Q5Evaluate ∫(sin x + sec²x) dx.
8Recap Recap
- Indefinite integration reverses differentiation: ∫f(x)dx is the family of all F with F'(x) = f(x).
- The constant C corresponds to the family of curves; use a condition to find C when given, otherwise keep C — write plus C the moment you integrate.
- Power rule: raise the exponent by one and divide by the new power; reciprocals give ln x; eˣ, sin, cos and sec² each have a matching formula.
- Linearity handles term by term with a single C; substitution in five moves: set u, find du, substitute, integrate, reverse-substitute.
- The standard check for any integral: differentiate the answer and see whether the integrand returns.