Integration is one of the two fundamental operations in calculus, alongside differentiation. While derivatives measure instantaneous rates of change, integrals accumulate quantities — computing areas under curves, volumes of solids, total distance from velocity, and accumulated quantities from rates. An indefinite integral (antiderivative) reverses differentiation: the integral of 2x is x² + C, where C is the constant of integration. A definite integral evaluates the net signed area between a function and the x-axis over a specific interval. The Fundamental Theorem of Calculus connects the two, stating that the definite integral from a to b equals F(b) minus F(a), where F is any antiderivative of the integrand. Integration techniques range from straightforward power rule applications to sophisticated methods like substitution, integration by parts, partial fractions, and trigonometric identities. Some functions — like e^(-x²), central to the normal distribution in statistics — have no closed-form antiderivative at all, requiring numerical methods such as Simpson's rule or Gaussian quadrature. Whether you are a student working through a calculus course, an engineer computing work or flux, or a physicist evaluating probability amplitudes, a solid grasp of integration is indispensable.
What is an Integral Calculator?
An integral calculator computes the antiderivative (indefinite integral) or definite integral of a mathematical function. For indefinite integrals, it finds F(x)+C where F'(x)=f(x). For definite integrals, it evaluates the net signed area under the curve f(x) between two bounds using the Fundamental Theorem of Calculus.
Supported Functions and Notation
This calculator supports: polynomials (x^2, 3*x^4), trigonometric functions (sin(x), cos(x), tan(x), sec(x), csc(x), cot(x)), exponentials (exp(x) or e^x), natural logarithm (ln(x)), square root (sqrt(x)), and composite functions. Use * for multiplication, ^ for exponentiation, and standard parentheses for grouping.