Integral Calculator
Calculate indefinite and definite integrals with step-by-step solutions
Understanding Integration
Integration is a fundamental concept in calculus that allows us to calculate areas, volumes, and accumulation of quantities over time or space. Whether you're studying physics, engineering, or mathematics, understanding integration is crucial for solving real-world problems.
Indefinite Integrals
An indefinite integral, or antiderivative, is a function whose derivative equals the integrand. The result includes a constant of integration (C) because derivatives of functions that differ only by a constant are equal.
Example:
∫x² dx = x³/3 + C
Definite Integrals
A definite integral calculates the signed area between a function and the x-axis over a specified interval. It's computed using the Fundamental Theorem of Calculus by evaluating the antiderivative at the bounds.
Example:
∫₁² x² dx = [x³/3]₁² = 7/3
Common Integration Rules
Power Rule
∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C
Constant Rule
∫a dx = ax + C
Sum Rule
∫(f(x) + g(x)) dx = ∫f(x) dx + ∫g(x) dx
Exponential Rule
∫eˣ dx = eˣ + C
Applications
Physics
- • Work and energy calculations
- • Center of mass determination
- • Fluid pressure and force
Engineering
- • Area calculations
- • Volume of revolution
- • Moment of inertia
Statistics
- • Probability distributions
- • Expected value calculations
- • Continuous random variables