Integral Calculator

Introduction

An integral is a core idea in calculus that helps you find the area under a curve. Think of it like adding up tiny slices of a shape to get the total area. Integrals are used to solve problems in math, science, and engineering every day. There are two main types: definite integrals, which give you a number (the exact area between two points), and indefinite integrals, which give you a function (the antiderivative). This integral calculator lets you type in a function and get the answer fast. It handles common functions like polynomials, trig functions, exponentials, and more. Whether you are a student learning calculus for the first time or just need a quick way to check your work, this tool makes solving integrals simple and easy.

How to Use Our Integral Calculator

Enter your function and limits of integration below. The calculator will compute the definite or indefinite integral and show you the result step by step.

Function: Type the math function you want to integrate. Use standard notation like x^2, sin(x), or e^x. This is the expression that will be integrated with respect to your chosen variable.

Variable: Enter the variable you are integrating with respect to. In most cases, this will be "x," but you can use any letter like "t" or "u."

Lower Limit: If you want to solve a definite integral, enter the lower bound of integration here. Leave this blank if you want an indefinite integral (general antiderivative).

Upper Limit: If you want to solve a definite integral, enter the upper bound of integration here. Leave this blank if you want an indefinite integral.

Understanding Integrals

An integral is a fundamental concept in calculus that lets you find the total amount of something when you know its rate of change. Think of it this way: if you know how fast a car is going at every moment, an integral tells you the total distance the car traveled. It's basically the reverse of taking a derivative.

Two Types of Integrals

There are two main types of integrals you should know about:

• Definite integrals give you an actual number. They calculate the total value between two specific points, called the lower limit and upper limit. Visually, a definite integral finds the area under a curve between those two points on a graph.
• Indefinite integrals give you a new function instead of a number. This new function is called the antiderivative. Since many different functions can have the same derivative, we always add a constant written as + C at the end.

How Integration Works

Integration follows a set of rules, just like addition or multiplication. Some common rules include:

• Power Rule: To integrate x raised to a power, you increase the power by 1 and divide by that new number. For example, the integral of x² is x³/3 + C. You can verify results involving exponents with dedicated tools.
• Sum Rule: You can split an integral of added terms into separate integrals for each term.
• Constant Rule: A number multiplied in front can be pulled outside the integral.

Why Integrals Matter

Integrals are used everywhere in science, engineering, and everyday math. They help calculate areas, volumes, average values, work done by a force, and even probabilities in statistics. For instance, in physics you can use integrals to determine quantities like kinetic energy, potential energy, or the displacement of an object when given its velocity function. In statistics, integrals underpin calculations like the z-score and confidence intervals. Anytime you need to add up a continuous quantity — something that doesn't come in neat, countable pieces — you need an integral.

The connection between derivatives and integrals is so important that it has its own name: the Fundamental Theorem of Calculus. This theorem says that differentiation and integration are opposite operations, much like how addition and subtraction undo each other. If you're also studying rates of change, our rate of change calculator is a helpful companion tool. For problems involving the geometry of curves — such as finding the length of a curve defined by an integral — try our arc length calculator. And if your integral involves logarithmic functions, our log calculator can help you simplify expressions before integrating.

Frequently Asked Questions

What is the difference between a definite and indefinite integral?

A definite integral has upper and lower bounds and gives you a number. It finds the exact area under a curve between two points. An indefinite integral has no bounds and gives you a function called the antiderivative. Indefinite integrals always include + C at the end because many functions can share the same derivative.

How do I enter my function into the calculator?

Type your function in the input box using standard math notation. Use * for multiplication, ^ for powers, and parentheses to group terms. For example, type x^2 for x squared, sin(x) for sine of x, or x*cos(x) for x times cosine of x. You can also click the Show Math Keyboard button to use the virtual keyboard.

What does the + C mean in my answer?

The + C stands for the constant of integration. When you find an indefinite integral, there are many possible answers that differ by a constant number. Since we don't know what that constant is, we write + C to show it could be any number. Definite integrals do not have + C because the constants cancel out when you subtract the bounds.

Can this calculator solve improper integrals?

Yes. Click the ∫₀^∞ Improper button at the top. You can set bounds to infinity (∞) or negative infinity (-∞). The calculator uses numerical methods to estimate the value. If the integral diverges (goes to infinity), it will tell you the result is divergent.

What is a double or triple integral?

A double integral integrates a function over a two-dimensional region. A triple integral integrates over a three-dimensional region. They are used to find areas, volumes, and other quantities in higher dimensions. This calculator can compute double and triple integrals numerically when you set bounds for each variable.

How do I use the natural language input mode?

Click Natural Language in the input mode area. Then type your integral in plain English, like "integrate x^2 from 0 to 5" or "integral of sin(x) dx". The calculator will parse your words and set up the integral for you automatically.

What integration methods does this calculator use?

The calculator uses several methods including the power rule, sum/difference rule, constant multiple rule, u-substitution, and known formulas for trig, exponential, logarithmic, hyperbolic, and inverse trig functions. For definite integrals without a closed-form solution, it uses adaptive Simpson's rule for numerical approximation.

Can I see the steps to solve my integral?

Yes. After clicking Calculate Integral, look for the Show Step-by-Step Solution button below the result. Click it to see each step the calculator used to find the answer, including substitutions and rules applied.

What does the graph show?

The graph shows two curves. The solid line is your original function (the integrand). The dashed line is the antiderivative. For definite integrals, the shaded region between the bounds shows the area that the integral calculates.

How do I enter infinity as a bound?

Type inf, ∞, or infinity in the bound field for positive infinity. For negative infinity, type -inf or -∞. The calculator will treat these as improper integral limits.

Can I integrate with respect to a variable other than x?

Yes. For definite, improper, double, and triple integrals, use the Variable dropdown to pick dy, dz, dt, or du. For indefinite integrals, just type your function using the variable you want and the calculator will detect it.

Why does the calculator say 'No closed-form found'?

Some functions do not have an antiderivative that can be written with basic math symbols. For example, e^(x²) has no simple antiderivative. When this happens, the calculator cannot give a symbolic answer. If you set bounds, it will still give you a numerical approximation of the definite integral.

Can I enter pi or e in my expression?

Yes. Type pi for π (about 3.14159) and e for Euler's number (about 2.71828). You can also click the π and e buttons on the virtual math keyboard. These work in both the function input and the bound fields.

What is u-substitution?

U-substitution is a technique for solving integrals. You replace a complicated part of the function with a new variable u to make the integral simpler. For example, in ∫ x·sin(x²) dx, you let u = x², so du = 2x dx. This turns the integral into ½ ∫ sin(u) du, which is easy to solve. The calculator applies this method automatically when it fits.

How accurate are the numerical results?

The calculator uses adaptive Simpson's rule with a tolerance of about 10⁻¹⁰, so numerical results are typically accurate to 8 or more decimal places. For improper integrals or functions with sharp spikes, accuracy may be slightly lower. Results are shown with up to 10 significant digits.

Can I use this calculator on my phone?

Yes. The calculator is fully responsive and works on phones and tablets. The virtual math keyboard adjusts to smaller screens. You can tap buttons to enter symbols instead of typing them.