Integral Calculator

An integral is a fundamental concept in calculus that represents the accumulation of quantities. It is the inverse operation of differentiation. Types of Integrals: 1. Indefinite Integral (Antiderivative): Notation: ∫ f(x) dx = F(x) + C Where: • f(x) = integrand (function being integrated) • dx = differential (variable of integration) • F(x) = antiderivative (F'(x) = f(x)) • C = constant of integration (arbitrary constant) 2. Definite Integral: Notation: ∫[a to b] f(x) dx = F(b) - F(a) Where: • a = lower limit of integration • b = upper limit of integration • Result is a specific numerical value Geometric Interpretation: • Definite integral represents the signed area between the curve y = f(x) and the x-axis from x = a to x = b • Area above x-axis is positive, below is negative Fundamental Theorem of Calculus: Links differentiation and integration: 1. First Part: If F(x) = ∫[a to x] f(t) dt, then F'(x) = f(x) 2. Second Part: ∫[a to b] f(x) dx = F(b) - F(a), where F'(x) = f(x) Common Integration Techniques: • Power Rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1) • U-Substitution: For composite functions • Integration by Parts: ∫ u dv = uv - ∫ v du • Partial Fractions: For rational functions • Trigonometric Substitution: For expressions with √(a² - x²), √(a² + x²), √(x² - a²) Applications: Calculating areas, volumes, arc lengths, surface areas, center of mass, work done by a force, probability distributions, accumulation of quantities over time, and solving differential equations.