Integral Calculator

Category: Calculus

- April 01, 2025

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Calculate definite and indefinite integrals. Enter a function and integration limits to get step-by-step solutions.

Integration Information

Integration Type:

Variable:

Function Input

Function to Integrate:

Lower Limit:

Upper Limit:

Advanced Options

Decimal Places:

Show integration steps

Show function plot

About Integration

Integration is a way of adding slices to find the whole. It is used to find areas, volumes, central points, and many useful things. It is the reverse of differentiation.

Types of Integrals

Indefinite Integral: Represents a family of functions (antiderivatives) whose derivative equals the integrand.
Definite Integral: Gives the area under a curve between two specific points.

Common Integration Rules

\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) (for \(n \neq -1\))
\(\int \sin(x) \, dx = -\cos(x) + C\)
\(\int \cos(x) \, dx = \sin(x) + C\)
\(\int e^x \, dx = e^x + C\)
\(\int \frac{1}{x} \, dx = \ln|x| + C\)

Integration Techniques

Substitution Method: Used when the integrand can be rewritten in a simpler form with a new variable.
Integration by Parts: Used for products of functions, applying the formula \(\int u \, dv = uv - \int v \, du\).
Partial Fractions: Used for rational functions by breaking them into simpler fractions.
Trigonometric Substitution: Used when the integrand contains specific algebraic expressions.

What Is the Integral Calculator?

The Integral Calculator helps you solve integrals both indefinite (antiderivatives) and definite (numerical value over an interval). It also offers step-by-step explanations and visual graphs to make integration more understandable and intuitive.

Whether you're learning calculus or need a quick way to evaluate an integral, this integration tool is built for convenience and clarity. It’s a valuable resource for students, teachers, engineers, and anyone working with mathematical functions.

Key Integration Formula:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \ne -1 \]

Why Use This Integration Tool?

Find antiderivatives instantly for a wide range of functions.
Solve integrals online using a clean and simple interface.
Choose between definite and indefinite integration depending on your needs.
Visualize the function, its antiderivative, and the area under the curve.
Follow step-by-step solutions for better understanding of the integration process.
Set constants and variables for customized integration problems.

How to Use the Calculator

Choose your integration type: indefinite or definite.
Enter your function in the input box (e.g., x^2, sin(x), e^x).
For definite integrals, specify the upper and lower bounds.
Set the integration variable (default is x) and any constants if needed.
Select how many decimal places you'd like the result to show.
Check or uncheck options to show solution steps and graphs.
Click Calculate Integral to get the result.

Interactive Graph and Step-by-Step Solution

The calculator displays a live graph of the function and its antiderivative. For definite integrals, it shades the area under the curve, making it easier to interpret the result visually. You'll also see detailed solution steps, which help break down how the integral was calculated.

Common Functions and Presets

Quickly insert common mathematical functions using preset buttons, such as:

x²
sin(x)
eˣ
ln(x)
1/x

Who Can Benefit from This Calculator?

This integration tool is useful for:

Students needing to compute indefinite integrals or evaluate definite integrals.
Teachers explaining the antiderivative steps in class.
Professionals looking to solve integrals as part of larger calculations.

It's also a great companion alongside tools like the Partial Derivative Calculator, Antiderivative Calculator, and Second Derivative Tool for deeper calculus analysis.

Frequently Asked Questions (FAQ)

Can I find both definite and indefinite integrals?
Yes, the calculator supports both. Just select your preferred option from the dropdown menu.

What functions are supported?
You can enter polynomials, trigonometric functions (like sin(x)), exponentials (like e^x), logarithms, and more.

Does it show solution steps?
Yes, you can enable step-by-step solutions to learn how the integral is solved.

Can I use it to learn?
Definitely. It’s perfect for understanding concepts like antiderivatives, the Fundamental Theorem of Calculus, and integration techniques.

Is graphing included?
Yes. It graphs the original function, its antiderivative, and the shaded area (for definite integrals).

Explore More Calculus Tools

If you're working with more advanced problems, try using these related calculators:

Partial Derivative Solver – for multivariable differentiation.
Derivative Calculator – to find first or higher-order derivatives instantly.
Second Derivative Tool – for concavity and inflection point analysis.
Antiderivative Calculator – to break down the process of finding antiderivatives.
Limit Solver – to find the limit of a function as it approaches a specific value.

This Integral Calculator helps simplify and visualize calculus problems. Whether you're trying to find the area under a curve, understand an antiderivative, or just solve integrals quickly, this tool is built to make your work easier and more accurate.