Supported formats:
- x^2 for x²
- 3x^2+2x-1 for polynomials
- sin(x), cos(x), tan(x)
- e^x or exp(x)
- ln(x) for natural log
- sqrt(x) for square root
Indefinite Integral
Finds the antiderivative F(x) such that F'(x) = f(x). Result includes + C for the constant of integration.
Definite Integral
Calculates the exact area under the curve from x = a to x = b using the Fundamental Theorem of Calculus: F(b) - F(a).
Integration is one of the two fundamental operations in calculus, alongside differentiation. While differentiation finds the rate of change of a function, integration performs the reverse operation - it finds the original function given its derivative. This process is also known as finding the antiderivative or primitive function. Integration has countless applications in mathematics, physics, engineering, and economics, from calculating areas and volumes to solving differential equations.
The integral symbol ∫ was introduced by Leibniz and is an elongated "S" representing "summa" (sum). This reflects the geometric interpretation of integration as the limit of a sum of infinitesimally small rectangles under a curve. The constant C in indefinite integrals represents the family of all antiderivatives, since any constant disappears when differentiated.
Several techniques exist for evaluating integrals depending on the form of the integrand. The power rule is the most basic, stating that ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ -1. Substitution (u-substitution) is used when the integrand contains a composite function, while integration by parts handles products of functions using the formula ∫u dv = uv - ∫v du.
More advanced techniques include partial fractions for rational functions, trigonometric substitution for integrands containing square roots of quadratic expressions, and trigonometric identities for products of trigonometric functions. Some integrals cannot be expressed in terms of elementary functions and require numerical methods or special functions.
This calculator uses symbolic integration rules for common function types. Results depend on correct function input and supported formats. For complex integrals involving advanced techniques like integration by parts or substitution, consider using specialized computer algebra systems. Always verify results for academic or professional work.