Integral Calculator

Free integral calculator with steps: indefinite and definite integrals by substitution, integration by parts, partial fractions and trig rules, improper integrals with infinite bounds, numeric values when there is no closed form, and a plot of the area.

∫ dx
Press Enter to run. Steps, a check by differentiation and a plot come with every answer.

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Free online integral calculator with steps

Type a function such as x e^x, 1/(x^2 - 1), sin^2 x cos x or sqrt(1 - x^2) to get its antiderivative (with + C) and a step-by-step solution: the constant and power rules, integrating term by term, u-substitution, integration by parts (also the "by parts twice" trick for e^x sin x), partial fractions, trigonometric rewrites and substitutions, completing the square and the standard arctangent, arcsine and logarithm forms.

  • Definite integrals: switch to Definite and enter the bounds — numbers, pi, expressions like pi/4 or sqrt(2), and -oo / oo (or ∞) for improper integrals. The answer shows F(b) − F(a) and its decimal value.
  • No closed form? Integrals like ∫₀¹ xˣ dx are evaluated numerically (mpmath's adaptive quadrature) to 15 significant digits.
  • Verified: the antiderivative is differentiated again and compared with your function.
  • A plot of the integrand with the area shaded, and a link to the 2D graphing calculator.

How it works

Everything runs in your own browser with SymPy, the open-source computer algebra system, compiled to WebAssembly with Python (about 10 MB the first time, cached afterwards). The steps are SymPy's own integration rule tree (the same method a person would use), written out as numbered steps. When no such method applies, the result comes from SymPy's general algorithm (Risch, Meijer G) and the page says so honestly.

Tips

Variable
Usually found automatically; you can also end the input with dt, dθ…, or type it in the Variable box. Other letters are treated as constants.
Logarithms
The antiderivative of 1/x is shown as ln|x|. ln and log both mean the natural logarithm.
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