Symbolic Integral Calculator

Integration has two closely related meanings. As accumulation, it measures total quantity built from a changing rate, such as distance from velocity or area under a curve. As an antiderivative, it reverses differentiation by finding a function whose derivative is the integrand.

For indefinite integrals, the answer includes a constant of integration because many functions differ only by a constant while sharing the same derivative. This constant is not decoration; it represents missing initial-condition information.

Many symbolic integrals are built from simple rules. The power rule integrates x^n into x^(n+1)/(n+1) when n is not -1. Linearity allows sums and constant multiples to be handled term by term. These rules make polynomial integration straightforward.

The exception n = -1 gives the natural logarithm because the derivative of ln|x| is 1/x. This special case is one reason integration requires more pattern awareness than differentiation.

Differentiation follows mechanical rules that usually simplify expressions. Integration often requires recognizing structure: substitution, integration by parts, partial fractions, trigonometric identities, or special functions. Some elementary-looking functions do not have elementary antiderivatives.

This asymmetry matters in symbolic systems. A derivative can almost always be produced in familiar form. An integral may require a clever transformation, a non-elementary function, or a numerical method.

A definite integral evaluates accumulation over an interval. The fundamental theorem of calculus connects this to antiderivatives: find an antiderivative, subtract its values at the endpoints, and get the net accumulated quantity.

The sign of the integral depends on whether the function lies above or below the axis. Area and net accumulation are related but not always identical. In physical problems, units help: integrating velocity over time gives distance displacement units, while integrating power over time gives energy.