Integral Calculator

An integral calculator evaluates integrals of mathematical functions. Integration is an important part of calculus and is used to find accumulated quantities, areas, displacement, volumes, and other quantities from rates of change.

What Is an Integral?

An integral represents accumulation. In many applications, it can be used to find the area under a curve or recover a function from its derivative.

There are two main types of integrals: indefinite integrals and definite integrals.

Indefinite Integral

An indefinite integral finds a family of antiderivatives and includes a constant of integration:

∫ f(x) dx = F(x) + C

Here, F(x) is an antiderivative of f(x), and C is the constant of integration.

Basic Integration Rule

For a power of x, the power rule for integration is:

∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C

where n ≠ −1.

For example:

∫ x² dx = x³ ÷ 3 + C

To verify the result, differentiate x³/3 + C. Its derivative is x².

Definite Integral

A definite integral has an upper and lower limit:

∫_a^b f(x) dx

It produces a numerical result rather than a family of functions. The value represents the net signed area between the function and the x-axis over the specified interval.

Example of a Definite Integral

Consider:

First find the antiderivative:

∫ x² dx = x³/3

Then evaluate it at the upper and lower limits:

[x³/3]₀² = 8/3 − 0 = 8/3

Therefore, the definite integral is approximately 2.667.

Common Integrals

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

Integration and Area

A definite integral can be used to calculate the net area between a curve and the x-axis. Areas below the x-axis contribute negative values to the integral, so a geometric total area may require additional calculations when the function changes sign.

Why Use an Integral Calculator?

Integrating complicated expressions by hand can involve several rules and algebraic steps. An integral calculator can quickly evaluate many common integrals and help verify manually calculated results.

Common Mistakes to Avoid

For indefinite integrals, remember to include the constant of integration C. For definite integrals, use the correct upper and lower limits and evaluate the antiderivative at both endpoints.

Also distinguish between net signed area and total geometric area when interpreting a definite integral.

Quick Summary

Integration is the reverse process of differentiation and is used to calculate accumulated quantities. An indefinite integral produces an antiderivative plus C, while a definite integral evaluates the accumulated net change over a specified interval.

Loading...