Integral Calculator

What Is an Integral?

An integral measures accumulation. Geometrically, the integral of a function f(x) gives the area between its curve and the x-axis over some interval — add up infinitely many infinitely thin rectangles under the curve and the total is the integral. Algebraically, integration is the reverse of differentiation: to integrate f(x) you look for a function F(x) whose derivative is f(x). That F(x) is called the antiderivative, and finding it is the whole job.

The notation ∫ f(x) dx reads "the integral of f of x with respect to x." The elongated S (∫) stands for sum, f(x) is the integrand (the function being integrated), and dx names the variable you are integrating over. Because integration reverses differentiation, it is the second pillar of calculus alongside the derivative calculator, which does the forward operation of finding slopes.

This calculator focuses on polynomials — expressions built from terms like 6x², 4x, and constants. Type a polynomial such as 6x^2 + 4x - 5 and it returns the antiderivative term by term, with the constant of integration and a full step-by-step solution. It does not handle trigonometric, exponential, logarithmic, or other non-polynomial functions; for those you would need substitution, integration by parts, or a full computer-algebra system.

The Power Rule for Integration

Every polynomial integral this tool computes comes from one rule: the reverse power rule (sometimes called the power rule for integration). To integrate a power of x, raise the exponent by one and divide by the new exponent.

With a coefficient in front, the constant simply rides along: ∫ c·xⁿ dx = c·xⁿ⁺¹/(n+1) + C. Two pieces of this rule do a lot of work:

• The constant multiple rule: a number multiplying the term stays out front, so ∫ 6x² dx = 6·(x³/3) = 2x³.
• The sum rule: a polynomial is integrated one term at a time, then the results are added. ∫ (f + g) dx = ∫ f dx + ∫ g dx.

A constant term is just x⁰, so the rule still applies: ∫ 5 dx = 5x. The only exponent the power rule cannot handle is n = −1, because dividing by n + 1 = 0 is undefined — that case (1/x) integrates to ln|x| instead, which is why a 1/x term falls outside this polynomial tool. If you need that logarithm, the logarithm calculator is the right companion.

Definite vs. Indefinite Integrals

Integrals come in two flavors, and the difference is whether you have bounds.

This calculator computes the indefinite integral — the antiderivative F(x) + C. That antiderivative is exactly the ingredient you need to evaluate a definite integral by hand: find F(x) here, then plug in the bounds and subtract, as shown in the worked example below.

The Constant of Integration (+C)

Why does every indefinite integral end in + C? Because the derivative of any constant is zero. The functions x³, x³ + 7, and x³ − 100 all have the same derivative, 3x², so when you reverse the process you cannot tell which constant was there originally. Writing + C honestly captures that whole family of antiderivatives at once. It is not optional decoration — leaving it off means you have written down only one of infinitely many correct answers.

How to Use This Integral Calculator

1. Type your polynomial. Enter f(x) in the input box using ^ for exponents — for example 6x^2 + 4x - 5 or x^3 - x. You can also tap a built-in example to load it instantly.
2. Press Calculate. The calculator integrates term by term with the reverse power rule and displays the antiderivative, including + C.
3. Read the step-by-step solution. Open the Steps tab to see each term raised and divided, the Visual tab to see the area under the curve, and Why it works for the reasoning.
4. Verify and practice. Use the built-in derivative check (it differentiates the answer back to f(x)), then try the Practice problems to test yourself.

Use ^ for any power, a plain number for a constant term, and + or - to separate terms. Here is the input syntax at a glance:

Worked Example 1 — Indefinite Integral

Let us integrate the polynomial ∫ (6x² + 4x − 5) dx term by term.

1. Integrate 6x². Raise the power: 2 + 1 = 3. Divide by the new power: 6/3 = 2. This term becomes 2x³.
2. Integrate 4x. Here x is x¹. Raise the power: 1 + 1 = 2. Divide by the new power: 4/2 = 2. This term becomes 2x².
3. Integrate −5. A constant is −5·x⁰, so it integrates to −5x.
4. Add the constant of integration. Combine the terms and append + C: 2x³ + 2x² − 5x + C.

Check it by differentiating: d/dx (2x³ + 2x² − 5x + C) = 6x² + 4x − 5, the original integrand. The answer is confirmed.

Worked Example 2 — Definite Integral with Limits

A definite integral reuses the same antiderivative, then evaluates it at the bounds. Let us compute ∫₀³ (2x + 1) dx, the signed area under y = 2x + 1 from x = 0 to x = 3.

1. Find the antiderivative. Integrate term by term: ∫ 2x dx = x², and ∫ 1 dx = x. So F(x) = x² + x. (No + C is needed — it cancels in the next step.)
2. Apply the Fundamental Theorem of Calculus. Evaluate F at the upper bound and subtract F at the lower bound: ∫₀³ (2x + 1) dx = F(3) − F(0).
3. Plug in the upper bound. F(3) = 3² + 3 = 9 + 3 = 12.
4. Plug in the lower bound. F(0) = 0² + 0 = 0.
5. Subtract. 12 − 0 = 12. The signed area under the curve between 0 and 3 is 12.

This calculator produces the F(x) in Step 1 for any polynomial; from there, evaluating the bounds and subtracting is the by-hand part shown above.

Integration and Differentiation: The Fundamental Theorem

Integration and differentiation are inverse operations, and the link between them is the Fundamental Theorem of Calculus (FTC). It comes in two parts. The first part says that integrating a function and then differentiating the result gives you back the original function — they undo each other. The second part is the rule you used in Example 2: if F is any antiderivative of f, then

This is the bridge between the two halves of calculus. The derivative calculator goes one direction (function → slope); this integral calculator goes the other (function → accumulated area). Because they are inverses, the surest way to verify any integral is to differentiate it: if you get the original integrand back, the integral is correct. The idea generalizes well beyond areas — the same accumulation logic underlies Riemann sums and the partial sums you can explore with the sequence and series calculator.

Common Mistakes to Avoid

Frequently Asked Questions

• What is an integral in simple terms?

  An integral is a way of adding things up. It measures the area between a curve and the x-axis, and it reverses differentiation by finding a function whose slope (derivative) is the one you started with. That function is called the antiderivative.

• What is the difference between a definite and an indefinite integral?

  An indefinite integral has no bounds and gives a function plus the constant of integration (+ C). A definite integral has a lower and upper limit and gives a single number — the signed area under the curve between those two limits, found with the Fundamental Theorem of Calculus.

• Why do we add + C to indefinite integrals?

  Because the derivative of any constant is zero. Functions like x³, x³ + 7, and x³ − 5 all share the derivative 3x², so reversing the process cannot recover the original constant. Writing + C represents every possible antiderivative at once.

• How do I integrate a polynomial term by term?

  Apply the reverse power rule to each term: raise its exponent by one and divide by the new exponent, keeping any coefficient out front. Integrate every term separately, add the results together, and finish with + C for an indefinite integral.

• Can every function be integrated?

  In theory most continuous functions have an antiderivative, but not every one can be written with elementary functions. Integrals like ∫ e^(−x²) dx have no formula in terms of standard functions and must be evaluated numerically. Polynomials, however, always integrate cleanly with the power rule.

• Does this integral calculator show the steps for free?

  Yes. This integral calculator is completely free with no login or paywall. For any polynomial you enter it shows the full step-by-step solution, the antiderivative with + C, a visual of the area under the curve, and a built-in check that differentiates the answer back to your original function.