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Integral Calculator with steps

Get an antiderivative that has been differentiated back to your function, or a definite integral evaluated as F(b) − F(a).

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Answer

Enter your numbers or pick an example — the answer and the steps appear here instantly.

Indefinite and definite integrals

  • Antiderivatives with + C

    Sums are split term by term, each with its method hint, and the result gets its + C. Differentiating the answer must return your function before it is shown.

  • Definite integrals between limits

    Enter a lower and an upper limit and the Fundamental Theorem gives F(b) − F(a), exact when possible plus a decimal.

  • Numerical fallback

    Some functions, such as sin(x²), have no elementary antiderivative. For those the definite value is computed with Simpson's rule and marked ≈.

  • Refuses impossible ranges

    If the function is undefined or unbounded between the limits, like 1/x from −1 to 1, you get an explanation instead of a misleading number.

How to integrate with the calculator

  1. Pick indefinite or definite

    Choose an antiderivative, or switch to definite when you need a number between two limits.

  2. Type the function and variable

    For example 3x^2 + 2x, x*sin(x) or 1/(1 + x^2), integrated with respect to x or another letter.

  3. Add limits if needed

    The lower limit goes in the first box and the upper in the second; swapping them flips the sign of the answer.

  4. Read the result and the outline

    Each term shows its method: power rule, constant multiple, substitution, integration by parts or a standard table result.

How to integrate by hand: the methods to know

Most first-course integrals use the power rule in reverse: ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C for any n ≠ −1, while ∫ 1/x dx = ln|x| + C covers the missing case. Integrals of sums split term by term and constants slide outside, so ∫ (3x² + 2x) dx = x³ + x² + C.

Substitution undoes the chain rule. In ∫ 2x·cos(x²) dx, let u = x², so du = 2x dx and the integral becomes ∫ cos u du = sin u + C = sin(x²) + C. Look for an inner function whose derivative also appears as a factor.

Integration by parts undoes the product rule: ∫ u dv = uv − ∫ v du. For ∫ x·sin x dx, take u = x and dv = sin x dx, so du = dx and v = −cos x. The result is −x·cos x + ∫ cos x dx = −x·cos x + sin x + C.

For a definite integral, find any antiderivative F and compute F(b) − F(a); the constant cancels. ∫ from 0 to 2 of 3x² dx = 2³ − 0³ = 8. Make sure the function is continuous on the whole interval first: for 1/x² from −1 to 1 the formula would give −2, which is impossible for a positive function, because the integral actually diverges at 0.

Frequently asked questions

Every function has infinitely many antiderivatives that differ by a constant, since the derivative of a constant is zero. The + C stands for all of them at once.

Functions like sin(x²) or e^(−x²) have no antiderivative built from elementary functions. For a definite integral the calculator then switches to Simpson's rule and labels the value ≈.

Because the function blows up or is undefined somewhere between your limits, like 1/x on −1 to 1. Applying F(b) − F(a) there would print a number that is simply wrong, so the calculator explains the problem instead.

It is differentiated back and compared with your original function. Only a result that passes appears on the answer card.

Yes; the integral, the term-by-term outline and the check all run locally in your browser with no account. The optional AI explanation is included in the Pro plans* (*fair use: up to 300 runs a day across the writing, study and code tools) or costs 2 credits without a plan, and signed-out visitors get a free preview of its first part.

Treat them as a way to learn the method and check your own work, and follow your course's rules on AI tools. The AI-written explanation can include mistakes, so compare it with the calculator's verified result.

Related calculus help

Ready to integrate?

Enter the function, add limits if you need a number, and see the method for each term.

Evaluate an integral
  • Checked by differentiation
  • Exact values plus decimals
  • Simpson's rule fallback