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Integral Calculator

Free web tool: Integral Calculator

Powers · Roots

Trigonometric

Exp · Log · Hyperbolic

Numbers · Operators

Example formulas

∫ f(x) dx =

1x^3 + 2sin(x) - e^x + 5x + C

Graph

f(x) F(x) = ∫f(x)dx

Steps

  1. ∫ 3x^2 dx = 3/3 x^3 = 1x^3
  2. ∫ 2cos(x) dx = 2sin(x)
  3. ∫ -e^x dx = -e^x
  4. ∫ 5 dx = 5x

Integration Rules

∫ c dx = cx + C
∫ x^n dx = x^(n+1)/(n+1) + C
∫ sin(x) dx = -cos(x) + C
∫ cos(x) dx = sin(x) + C
∫ tan(x) dx = -ln|cos(x)| + C
∫ cot(x) dx = ln|sin(x)| + C
∫ sec²(x) dx = tan(x) + C
∫ csc²(x) dx = -cot(x) + C
∫ e^x dx = e^x + C
∫ 1/x dx = ln|x| + C
∫ sinh(x) dx = cosh(x) + C
∫ cosh(x) dx = sinh(x) + C

About Integral Calculator

The Integral Calculator is a free, browser-based tool that computes antiderivatives and evaluates definite integrals for a wide range of common functions. It supports polynomial terms (x^n), trigonometric functions (sin, cos, sec²), the exponential function (e^x), and the natural logarithm (1/x). Type any combination of these terms into the input field and the calculator instantly shows the full antiderivative along with step-by-step integration rules applied to each term.

Students in calculus courses, engineers performing analytical work, and scientists verifying hand calculations will all find this tool valuable. The definite integral mode lets you specify a lower and upper bound and immediately computes the numerical value of the definite integral using the Fundamental Theorem of Calculus: F(b) − F(a). Every result is accompanied by the individual integration steps so you can follow exactly how each term was integrated.

All computation runs entirely inside your web browser using JavaScript. The parser breaks the input expression into individual terms, applies the correct symbolic integration rule to each term, combines the results into the antiderivative, and — when bounds are provided — evaluates F(b) − F(a) numerically. No data is ever sent to a server, which means your work remains private and the tool works offline once the page has loaded.

Key Features

  • Symbolic antiderivative computation: x^n → x^(n+1)/(n+1), sin(x) → -cos(x), cos(x) → sin(x), e^x → e^x, 1/x → ln|x|, sec²(x) → tan(x)
  • Definite integral evaluation using F(b) − F(a) with user-specified lower and upper bounds
  • Step-by-step integration breakdown showing each term's rule and intermediate result
  • Parses multi-term expressions with any combination of supported function types and numeric coefficients
  • Real-time results: antiderivative updates instantly as you type without pressing any button
  • Built-in integration rules reference card displaying all seven supported rules at a glance
  • 100% client-side processing — no server, no sign-up, works offline after page load
  • Dark mode support and responsive layout optimized for desktop, tablet, and mobile

Frequently Asked Questions

What functions can the Integral Calculator integrate?

The calculator supports constants, polynomial terms (x^n for any real n), sin(x), cos(x), sec²(x), e^x, and 1/x (which integrates to ln|x|). You can combine any of these in a single expression using + and - operators with numeric coefficients.

How do I enter a multi-term function like 3x² + 2cos(x) − e^x?

Type the expression directly as "3x^2 + 2cos(x) - e^x". The parser handles mixed combinations. For the exponential, write e^x or exp. For natural log integration (1/x), write 1/x or /x.

How does the definite integral mode work?

Check the "Definite Integral" checkbox and enter your lower (a) and upper (b) bounds. The tool evaluates the antiderivative F at both points and returns F(b) − F(a), showing both intermediate values for verification.

Why does the result show "+ C"?

The constant of integration C is included because indefinite integrals are defined up to an arbitrary constant. Any function of the form F(x) + C has the same derivative f(x). For definite integrals the constant cancels out in F(b) − F(a).

What is the integration rule for x^(−1) or 1/x?

The integral of 1/x is ln|x| + C, which is a special case because the standard power rule x^n → x^(n+1)/(n+1) would produce division by zero when n = −1. The calculator handles this automatically.

Can I integrate x^(−2) or other negative powers?

Yes. Write it as x^-2. The calculator applies the power rule: x^-2 integrates to x^(-1)/(-1) = -x^(-1) = -1/x. For the special case x^-1 (or 1/x) it automatically uses the ln|x| rule.

Is the Integral Calculator accurate for fractional exponents?

Yes. For example, x^0.5 (square root of x) integrates to (2/3)x^1.5. The calculator computes the new coefficient x^(n+1)/(n+1) in full floating-point precision and rounds only for display.

Does the calculator support integration by substitution or by parts?

The current version applies direct integration rules term by term. It does not perform symbolic substitution or integration by parts. For composite functions such as sin(2x) or e^(3x), you should apply the substitution mentally and adjust the coefficient before entering the expression.