Integral Calculator with Step-by-Step Solutions

This integral calculator computes both indefinite and definite integrals and shows the full working behind every result. Type a function, pick a mode, and read the antiderivative, the numeric value, and the rule applied at each step.

An integral is the reverse of a derivative — the antiderivative of a function. An indefinite integral returns a general formula plus a constant C, while a definite integral evaluates that formula between a lower and upper bound to give an exact area-under-the-curve value. The tool is built for students learning calculus, teachers preparing examples, and anyone wanting to check a hand-worked integral.

Private by design: every integral is parsed and solved right in your browser. Nothing you type is ever uploaded to a server.

How to Use the Integral Calculator

Choose Indefinite or Definite

Use the tabs at the top to pick Indefinite for a general antiderivative (the result ends in + C) or Definite to compute a numeric value between two bounds.

Enter your function

Type the function in standard notation — for example x^2, sin(x), e^x, ln(x), sqrt(x), or a combination like 2*x^2 + 3*x. Tap a button in the Common Functions palette to insert the right syntax instantly.

Set the bounds (definite only)

In Definite mode, enter the lower bound (a) and upper bound (b). You can use plain numbers or constants such as pi and e, and even expressions like 2*pi.

Read the result and steps

The result updates as you type: the antiderivative formula, the numeric value for definite integrals, and a collapsible step-by-step breakdown naming the integration rule used. Need a different letter? Switch the Variable selector between x, t, u, and θ.

Features

Indefinite & Definite Modes

Find a general antiderivative with + C, or evaluate a definite integral between bounds a and b for an exact value.

Wide Function Support

Integrates polynomials, roots and 1/x, trigonometric and inverse-trig functions, plus exponential and logarithmic functions.

Multiple Integration Rules

Applies the power, constant, sum/difference, constant-multiple, trigonometric, exponential, and logarithm rules — and integration by parts for products like x·sin(x).

Step-by-Step Solutions

Each answer expands into numbered steps showing the rule applied and the formula behind it, so you can follow the working.

Real-Time Preview

A formatted integral preview updates as you type, with bounds shown on the integral sign so you can confirm your input before reading the result.

Function Palette & Examples

One-tap buttons insert common functions with correct syntax, and ready-made examples load a function and bounds for you instantly.

Switchable Variable

Integrate with respect to x, t, u, or θ — useful for problems that use a variable other than x.

Runs in Your Browser

All parsing and integration happen locally — no sign-up, no uploads, and instant results every time.

Frequently Asked Questions

How do I enter a function to integrate?

Type the function in the input box using standard notation. Use ^ for exponents (x^2, x^3), sqrt(x) for a square root, and named functions such as sin(x), e^x, or ln(x). You can also tap a button in the Common Functions palette to insert the correct syntax.

What is the difference between a definite and indefinite integral?

An indefinite integral gives the general antiderivative — a formula plus a constant C. A definite integral takes that formula and evaluates it between a lower bound (a) and upper bound (b), returning a single numeric value that represents the area under the curve.

Does it show step-by-step working?

Yes. Every result includes a collapsible Step-by-step Solution that lists each step in order, names the integration rule applied (such as the power rule or integration by parts), and shows the formula used.

Why does the indefinite integral include "+ C"?

The constant of integration C appears because the derivative of any constant is zero. That means infinitely many functions share the same derivative, differing only by a constant — so the indefinite integral represents that whole family with a trailing + C.

Can it do integration by parts and other rules?

Yes. The calculator applies the power, constant, sum/difference, constant-multiple, trigonometric, exponential, and logarithm rules automatically. It also uses integration by parts for products such as x·sin(x), x·cos(x), and x·e^x.

Can I use pi or e in the bounds?

Yes. In the bound fields you can enter pi for π (about 3.14159) and e for Euler's number (about 2.71828), as well as simple expressions like 2*pi or 1/2.

Why can't it integrate my function?

The calculator uses rule-based symbolic integration, so it covers standard functions and common combinations rather than every possible expression. Some functions have no elementary antiderivative, and very complex inputs may fall outside the supported rules. Try simplifying the expression or splitting it into terms it recognizes.

How accurate are the definite integral results?

Definite integrals are evaluated from the exact antiderivative and the numeric value is rounded to about six decimal places, which is precise enough for typical homework and practical use.