Constant of integration

A constant added to an indefinite integral because differentiation erases additive constants. It represents the family of antiderivatives of a function on an interval.

Antiderivative: A function whose derivative equals a given function on an interval. The constant distinguishes the full family of antiderivatives that share one derivative.
Derivative: A function's instantaneous rate of change with respect to its input. The derivative is the operation that cannot detect an added constant.
Separation of variables: A method for solving differential equations by placing each variable's terms on opposite sides before integration. Integrating separated sides introduces constants that are combined into a solution parameter.
Constant of proportionality: A fixed factor relating two quantities that vary in direct proportion. It multiplies a quantity, whereas an integration constant is added to an antiderivative.
Indefinite integral: Notation for the family of antiderivatives of a function, conventionally written with an integration constant. Its notation explicitly records the arbitrary additive constant.
Power rule: A differentiation rule stating that the derivative of x raised to n is n times x raised to n minus one. Its reverse operation illustrates why an antiderivative requires an additive constant.
Separable differential equation: A first-order differential equation whose variables can be separated into a product of single-variable factors. Its integrated general solution contains a constant later fixed by data.
Constant of integration in probability: An additive constant sometimes introduced when integrating a probability density to obtain a cumulative distribution function. Normalization of a cumulative distribution fixes the additive constant through its boundary value.
Differentiation: The process of finding a function's derivative, measuring its local rate of change. Differentiation maps every member of an additive family of antiderivatives to the same function.
Linearity of differentiation: The property that differentiation preserves sums and constant multiples of functions. The derivative of a sum ignores any constant term because constants differentiate to zero.

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