Integral Calculator

How to Use the Integral Calculator

1. Enter your function: Type the function you want to integrate using x as the variable. Use standard mathematical notation with operators like ^, *, /, +, and -.
2. Click Calculate: The calculator will compute the indefinite integral and display the result with the constant of integration (+ C).
3. View the steps: See the integration rule applied and the step-by-step solution to understand how the integral was calculated.
4. Try different functions: Experiment with polynomials, trigonometric functions, exponentials, and logarithmic functions to see how integration works.

What is Integration?

Integration is the reverse process of differentiation in calculus. It's used to find the original function when given its derivative, or to calculate areas under curves, volumes, and accumulated quantities. An indefinite integral represents a family of functions that differ by a constant, denoted by "+ C".

Integration is fundamental in physics, engineering, economics, and statistics for solving problems involving accumulation, area, volume, center of mass, work, and probability distributions.

Basic Integration Rules

Power Rule: The integral of x^n is (x^(n+1))/(n+1) + C, where n is not equal to -1. This is the most commonly used integration rule for polynomial functions.

Exponential Rule: The integral of e^x is e^x + C. The exponential function is unique in that it's its own derivative and integral.

Logarithmic Rule: The integral of 1/x is ln|x| + C. This is a special case not covered by the power rule.

Trigonometric Functions: The integral of sin(x) is -cos(x) + C, and the integral of cos(x) is sin(x) + C. These follow from the derivatives of trigonometric functions.

Common Integration Applications

Area Under Curves: Definite integrals calculate the exact area between a function and the x-axis over a specified interval, useful in geometry and probability.

Physics and Motion: Integrating acceleration gives velocity, and integrating velocity gives position. This is essential for analyzing motion in mechanics.

Economics: Integration helps find total cost from marginal cost, total revenue from marginal revenue, and consumer surplus from demand functions.

Probability: Probability density functions are integrated to find cumulative probabilities and expected values in statistics.

Supported Functions

This calculator supports power functions (x^n), exponential functions (e^x), trigonometric functions (sin(x), cos(x)), logarithmic functions (1/x), and linear combinations of these basic functions. Use proper mathematical notation and parentheses for complex expressions.

Differentiation is the inverse of integration - use the Derivative Calculator to find derivatives. The Limit Calculator helps with understanding improper integrals.