Integral calculator, with every step

Add bounds for a definite integral.

Type a function and get the exact answer, the rule used at every step, and a graph. It handles integrals, derivatives and limits, runs in your browser, and is free with no sign-up.

How to use the calculator, step by step

1. Choose Integral, Derivative or Limit.
2. Type your function in the Function f(x) box. Press Try one to see an example.
3. Open Options to set the variable, the limits of integration, the order of the derivative or the point a limit approaches.
4. Press Work it out, then read the working. Use Copy the working or Save as text to keep it.

Each answer comes with the rule used at every step, written out in full, and a graph of the function next to its derivative, its antiderivative or the point the limit approaches. Everything is worked out in your browser.

Integrals

Indefinite integrals use the power rule, the standard table of exponential, logarithm and trigonometric integrals, substitution, integration by parts, trigonometric identities and partial fractions. Every antiderivative is checked by differentiating it again before it is shown. Some functions, such as e^x² or sin(x)/x, have no antiderivative made of ordinary functions; you are told so plainly. Add bounds, including inf, for a definite integral: the exact value is given when one exists, otherwise a numeric value, and improper integrals that blow up are reported as divergent.

Derivatives

Derivatives show the sum, constant multiple, power, product, quotient and chain rules one at a time, along with the derivatives of exponentials, logarithms, trigonometric and inverse trigonometric functions. Choose a higher order for second or third derivatives, or give a value to evaluate the derivative at a point.

Limits

Limits start with direct substitution. When that gives 0/0 or ∞/∞, the working factors and cancels, compares the highest powers, multiplies by a conjugate or applies L’Hôpital’s rule. Forms like 0·∞, ∞−∞ and 1^∞ are rewritten first. One-sided limits are supported, and a two-sided limit whose sides disagree is reported as not existing.

Typing functions

Write powers with ^, like x^3 or e^(2x). Square roots are sqrt(x), natural logs are ln(x) or log(x), and log10(x) is base 10. Inverse functions are arcsin, arccos and arctan. Products can be written without a sign, so 2x sin(x) works. Other letters, such as a or k, are treated as constants in derivatives and integrals.

Using it to study

Work a problem on paper first, then compare your steps with the ones shown. If your answer differs, look for the first step where the rules diverge. To solve an equation or plot several functions, try the equation solver or the graphing calculator. Your input stays in the browser.

Questions

Why does it say there is no antiderivative?

Some functions, such as e^(x^2), cannot be integrated into ordinary functions. A definite integral of them can still be given as a numeric value.

Does it add the constant of integration?

Indefinite integrals are antiderivatives, so remember to add + C when you write the answer.

Can I find a limit from one side only?

Yes. Set Direction to from the left or from the right.

Can I use it in a test?

It is meant for learning and checking your work. Follow your teacher's rules about calculators in tests.