Integral Calculator Step by Step Free

Integrals are fundamental in calculus for finding areas under curves, determining volumes, and solving differential equations. This guide explains how to use our free integral calculator to solve both definite and indefinite integrals step by step.

What is an Integral?

An integral represents the area under a curve between two points. It can be calculated as the limit of a Riemann sum. Integrals are used in physics, engineering, economics, and many other fields to model continuous quantities.

Key concept: The integral of a function f(x) with respect to x is the antiderivative F(x) such that F'(x) = f(x).

Types of Integrals

Indefinite Integrals

Indefinite integrals find the antiderivative of a function, represented with a constant of integration (C).

Definite Integrals

Definite integrals calculate the exact area under a curve between two limits, a and b.

Definite Integral Formula:

∫[a to b] f(x) dx = F(b) - F(a)

How to Calculate Integrals

Identify the type of integral (definite or indefinite).
Apply integration rules to find the antiderivative.
For definite integrals, evaluate the antiderivative at the upper and lower limits.
Subtract the lower limit evaluation from the upper limit evaluation.

Common Integral Formulas

Function	Integral
x^n	(x^(n+1))/(n+1) + C (n ≠ -1)
1/x	ln\|x\| + C
e^x	e^x + C
sin(x)	-cos(x) + C
cos(x)	sin(x) + C

Example Calculations

Example 1: Indefinite Integral

Find the integral of 3x².

∫3x² dx = x³ + C

Example 2: Definite Integral

Calculate the area under x² from 0 to 1.

∫[0 to 1] x² dx = (1³/3) - (0³/3) = 1/3

FAQ

What is the difference between definite and indefinite integrals?

Definite integrals calculate the exact area under a curve between two limits, while indefinite integrals find the general antiderivative with a constant of integration.

Can this calculator solve integrals with trigonometric functions?

Yes, our calculator can handle integrals involving sin(x), cos(x), and other trigonometric functions.

What if the integral doesn't match any of the common formulas?

For complex integrals, you may need to use integration techniques like substitution, integration by parts, or partial fractions.