Antiderivatives

A function F(x) is called an antiderivative of f(x) on an interval if F'(x) = f(x) for every x in that interval. For example, since the derivative of x² is 2x, we say x² is an antiderivative of 2x. Reversing the power rule: if f(x) = xⁿ (where n ≠ -1), then one antiderivative is x^(n+1)/(n+1).

Because d/dx [x² + 5] = 2x and d/dx [x² − 3] = 2x as well, there are infinitely many antiderivatives of 2x. They all differ by a constant. We express this family as x² + C, where C is any real number. This +C is the constant of integration. When you are given additional information (called an initial condition), you can solve for the specific value of C and obtain a particular antiderivative.

The indefinite integral ∫ f(x) dx is defined as the general antiderivative F(x) + C. The symbol ∫ is an elongated 'S' (from the Latin 'summa'), f(x) is called the integrand, and dx indicates that x is the variable of integration. The result is always a function (plus C), distinguishing it from the definite integral, which yields a number.

Several rules follow directly by reversing differentiation rules. Power Rule: ∫ xⁿ dx = x^(n+1)/(n+1) + C, for n ≠ -1. Constant Multiple Rule: ∫ k·f(x) dx = k·∫ f(x) dx. Sum Rule: ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx. Key results: ∫ cos(x) dx = sin(x) + C, ∫ sin(x) dx = −cos(x) + C, ∫ eˣ dx = eˣ + C, ∫ (1/x) dx = ln|x| + C.

When a problem specifies the value of the antiderivative at a particular point, you can determine C exactly. For example, if F'(x) = 3x² and F(0) = 7, first find the general antiderivative: F(x) = x³ + C. Then substitute the initial condition: F(0) = 0³ + C = 7, so C = 7. The particular solution is F(x) = x³ + 7. This technique is essential in physics when you know an object's initial position or velocity.

• \int — Integral sign (none): Indicates antidifferentiation; the elongated S stands for 'sum' and signals that we are finding the antiderivative.
• x^n — Integrand (none): The function being antidifferentiated. Here it is a power function with exponent n.
• dx — Differential (none): Indicates the variable of integration is x. It pairs with the integral sign to complete the notation.
• \frac{x^{n+1}}{n+1} — Antiderivative term (none): The result of applying the power rule: the exponent increases by one, and we divide by the new exponent.
• C — Constant of integration (none): An arbitrary real constant representing all possible vertical shifts of the antiderivative function. It encodes the non-uniqueness of antiderivatives.

Problem. A particle moves along a straight line with velocity v(t) = 6t² − 4t + 1 (in meters per second). At time t = 0, its position is s(0) = 3 meters. Find the position function s(t).

Answer. The position function is s(t) = 2t³ − 2t² + t + 3 meters.

• The Power Rule for Differentiation — The antiderivative power rule is the exact reverse of the differentiation power rule. Mastering d/dx[xⁿ] = nxⁿ⁻¹ makes the integration rule ∫xⁿ dx = xⁿ⁺¹/(n+1) + C intuitive and easy to verify.
• The Fundamental Theorem of Calculus — The Fundamental Theorem connects antiderivatives to definite integrals, showing that ∫ from a to b of f(x) dx = F(b) − F(a). This is why computing antiderivatives is the key technique for evaluating areas and accumulated change.
• Definite Integrals — Definite integrals compute numerical values (areas, displacements, totals) and rely directly on antiderivatives for their evaluation via the Fundamental Theorem.
• Initial Value Problems — Initial value problems use antiderivatives together with a known function value to determine the constant C and find a unique particular solution — a critical skill in physics and differential equations.