The Indefinite Integral and Fundamental Properties of Antiderivatives

An antiderivative of f is a function F with F′ = f, and the indefinite integral ∫ f(x) dx = F(x) + C is the family of all antiderivatives, where C is an arbitrary constant. In brief: ∫ x² dx = x³/3 + C, because the derivative of x³/3 + C is x² for every constant C. Integration reverses differentiation, and every integral can be checked by differentiating the answer.

If differentiation answers “how fast does it change?”, integration answers “what function changes at this rate?”. Given a velocity, the antiderivative gives the position; given the marginal cost, it gives the cost. This guide defines the antiderivative, explains why the constant C appears, lists the basic integrals and the properties (linearity), shows how to solve initial value problems and gives a table of integral formulas, with worked examples.

What is an antiderivative?

A function F is an antiderivative of f on an interval if F′(x) = f(x) for all x in the interval. If F is one antiderivative, then so is F + C for any constant C, since the derivative of a constant is 0. Conversely, if F and G are antiderivatives of the same f on an interval, then (F − G)′ = 0, so F − G is constant. So all antiderivatives of f differ by a constant, and we write

∫ f(x) dx = F(x) + C.

The symbol ∫ is a stretched S (for “sum”), f is the integrand and dx indicates the variable of integration. Every continuous function has an antiderivative, but for some functions, such as e^−x2 or sin x/x, the antiderivative cannot be written with elementary functions.

What are the fundamental properties?

• Reverse relation: (∫ f(x) dx)′ = f(x) and ∫ F′(x) dx = F(x) + C.
• Constant multiple: ∫ k f(x) dx = k ∫ f(x) dx.
• Sum and difference: ∫ [f(x) ± g(x)] dx = ∫ f dx ± ∫ g dx.
• No product or quotient rule: ∫ f g dx ≠ ∫ f dx · ∫ g dx. Products are handled by substitution and by parts.
• Linear inner function: if ∫ f(x) dx = F(x) + C, then ∫ f(ax + b) dx = F(ax + b)/a + C for a ≠ 0.

What are the basic integrals?

Each formula is obtained by reversing a derivative formula.

• ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, for n ≠ −1.
• ∫ 1/x dx = ln|x| + C.
• ∫ e^x dx = e^x + C, ∫ a^x dx = a^x/ln a + C.
• ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C.
• ∫ 1/cos²x dx = tan x + C.
• ∫ 1/(1 + x²) dx = arctan x + C, ∫ 1/√(1 − x²) dx = arcsin x + C.

The case n = −1 is the exception that leads to the natural logarithm: the absolute value is needed because 1/x is defined for negative x as well.

How do you find a particular antiderivative?

The constant C is determined by an extra condition, such as the value of F at a point (an initial condition). The problem “find F with F′ = f and F(a) = b” is an initial value problem.

Example 1

Find F with F′(x) = 6x² − 4x + 1 and F(1) = 5. F(x) = 2x³ − 2x² + x + C. F(1) = 2 − 2 + 1 + C = 1 + C = 5, so C = 4. F(x) = 2x³ − 2x² + x + 4.

Example 2: motion

A car has the acceleration a(t) = 6t m/s², the initial velocity 2 m/s and the initial position 0. v(t) = ∫ 6t dt = 3t² + C₁, and v(0) = 2 gives C₁ = 2. s(t) = ∫ (3t² + 2) dt = t³ + 2t + C₂, with s(0) = 0 giving C₂ = 0. After 3 s: v = 29 m/s and s = 33 m.

Example 3: marginal cost

The marginal cost is C′(q) = 0.06q + 4 dollars per unit and the fixed cost is 500. C(q) = 0.03q² + 4q + 500. For q = 100: 300 + 400 + 500 = 1,200 dollars.

Worked examples

Example 4: polynomials and roots

∫ (3x⁴ − 2x + 7) dx = 3x⁵/5 − x² + 7x + C. And ∫ √x dx = ∫ x^1/2 dx = (2/3)x^3/2 + C. And ∫ 1/x³ dx = ∫ x⁻³ dx = −1/(2x²) + C.

Example 5: rewriting the integrand

∫ (x + 1)²/x dx = ∫ (x² + 2x + 1)/x dx = ∫ (x + 2 + 1/x) dx = x²/2 + 2x + ln|x| + C. And ∫ (2x³ − 5x)/x dx = ∫ (2x² − 5) dx = 2x³/3 − 5x + C.

Example 6: linear inner function

∫ cos(3x) dx = sin(3x)/3 + C. ∫ e^−2x dx = −e^−2x/2 + C. ∫ (2x + 5)⁴ dx = (2x + 5)⁵/10 + C. ∫ 1/(4x − 1) dx = (1/4) ln|4x − 1| + C.

Example 7: trigonometric

∫ (sin x + 2 cos x) dx = −cos x + 2 sin x + C. And ∫ tan²x dx = ∫ (1/cos²x − 1) dx = tan x − x + C. And ∫ sin²x dx = ∫ (1 − cos 2x)/2 dx = x/2 − sin 2x/4 + C.

Example 8: check by differentiating

Claim: ∫ x e^x dx = (x − 1)e^x + C. Check: ((x − 1)e^x)′ = e^x + (x − 1)e^x = x e^x ✓.

Example 9: the family of solutions

The curves y = x³/3 + C all have slope x² at each x. They are vertical translates of each other. The one through (3, 0) has C = −9: y = x³/3 − 9.

Table of basic integrals

Where does the integral come from?

The idea of finding areas by summing small pieces goes back to Eudoxus and Archimedes. Newton and Leibniz in the seventeenth century recognised that finding an antiderivative is the inverse of differentiation, and used it to compute areas. The integral sign ∫ was introduced by Leibniz in 1675, from the long S of the Latin summa.

Where is it used?

Antiderivatives recover position from velocity, velocity from acceleration, total cost from marginal cost, charge from current and population from growth rate. Differential equations such as y′ = ky (growth and decay) are solved by integration: y′/y = k gives ln|y| = kt + C and y = Ae^kt. Many probability density functions are integrated to find probabilities.

Which mistakes are common?

Forgetting the constant C

The indefinite integral is a family of functions. Write + C (but not for definite integrals).

Using the power rule for n = −1

∫ x⁻¹ dx is ln|x| + C, not x⁰/0.

Integrating products term by term

∫ x · e^x dx is not (x²/2)(e^x). Use integration by parts.

Not checking the answer

Differentiate the result: you must get the integrand back.