Integral Calculator - Symbolic & Definite Integration

f(x)
x	f(x)
-5.0000	-235.0000
-3.8889	-127.0850
-2.7778	-57.8532
-1.6667	-19.0741
-0.5556	-2.5171
0.5556	0.0480
1.6667	-3.1481
2.7778	-3.8752
3.8889	6.0974
5.0000	35.0000

How to use this tool

Enter the function. Type your function of x in the input box. Use ^ for exponents, * for multiplication, and function names like sin, cos, exp, ln, sqrt.
Choose integral type. Leave the bounds empty for an indefinite integral (antiderivative + C), or enter lower and upper bounds for a definite integral.
Click ∫ dx. Click ∫ dx or press Enter to compute. The antiderivative and any definite value appear instantly.
Read the result. The antiderivative is shown in function notation. For definite integrals, the numerical value equals F(upper) − F(lower). Steps explain which rules were applied.

Integration rules quick reference

Rule	Formula
Power rule	∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
Reciprocal	∫1/x dx = ln\|x\| + C
Constant	∫c dx = cx + C
Sum/difference	∫(f±g) dx = ∫f dx ± ∫g dx
Constant multiple	∫c·f(x) dx = c·∫f(x) dx
Substitution	∫f(g(x))·g'(x) dx = F(g(x)) + C
Integration by parts	∫u dv = uv − ∫v du
sin	∫sin(x) dx = −cos(x) + C
cos	∫cos(x) dx = sin(x) + C
eˣ	∫eˣ dx = eˣ + C

Substitution and integration by parts are included as study references only. This calculator does not solve those techniques generally; it automatically applies a limited set of elementary symbolic rules. For definite integrals outside those rules, it can approximate the value numerically when the expression can be evaluated.

Numerical vs. symbolic integration

Not all integrals have a closed-form solution expressible in elementary functions. There are two complementary approaches:

Symbolic integration (what this calculator uses): finds an exact antiderivative formula, e.g. ∫3x² dx = x³ + C. Not all functions have closed-form antiderivatives (∫e^(x²) dx has none).
Numerical integration: approximates definite integrals using methods like Simpson's rule, Gaussian quadrature, or Monte Carlo integration. Useful when no closed form exists or when integrating experimental data.

Definite vs. indefinite integrals

A definite integral ∫ₐᵇ f(x) dx has two bounds and yields a number - the net signed area between the curve and the x-axis from a to b. An indefinite integral ∫f(x) dx has no bounds and yields a family of functions F(x) + C, where C is an arbitrary constant representing vertical shifts of the antiderivative.

What is integration?

Integration is the reverse of differentiation. Given a function f(x), finding its integral means finding a function F(x) whose derivative equals f(x). Integration also measures the area under a curve, one of the two pillars of calculus alongside differentiation.

Key integration rules

Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
Reciprocal rule: ∫x⁻¹ dx = ln|x| + C
Constant rule: ∫c dx = cx + C
Linearity: ∫(f ± g) dx = ∫f dx ± ∫g dx
Constant multiple: ∫c·f(x) dx = c·∫f(x) dx
sin: ∫sin(x) dx = −cos(x) + C
cos: ∫cos(x) dx = sin(x) + C
exp: ∫eˣ dx = eˣ + C
ln: ∫ln(x) dx = x·ln(x) − x + C

Fundamental Theorem of Calculus

If F is an antiderivative of f on [a, b], then ∫ₐᵇ f(x) dx = F(b) − F(a). This theorem connects differentiation and integration: the definite integral of f from a to b equals the net change in any antiderivative over that interval.

Input syntax

Write multiplication explicitly with *, such as 2*x; 2x is rejected as unexpected input. The parser accepts the functions below for numeric evaluation, while symbolic antiderivatives support a smaller set of forms.

Exponents: x^3 or x^(1/2)
Multiplication: * is required, as in 3*x^2
Numeric functions: sin(x), cos(x), tan(x), exp(x), ln(x), sqrt(x), abs(x). Symbolic forms are limited to supported elementary rules; tan and abs are numeric-only.
Combinations: 3*x^2 + sin(2*x) - exp(-x)

Frequently asked questions

What is an antiderivative?

An antiderivative (or indefinite integral) of a function f(x) is any function F(x) such that F′(x) = f(x). Because the derivative of any constant is zero, antiderivatives are only unique up to an additive constant C - written as ∫f(x) dx = F(x) + C.

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

An indefinite integral ∫f(x) dx gives a family of antiderivatives F(x) + C. A definite integral ∫ₐᵇ f(x) dx gives a specific number, equal to F(b) − F(a) by the Fundamental Theorem of Calculus. Geometrically it measures the net signed area under the curve from x = a to x = b.

Which functions can be integrated symbolically?

The calculator handles polynomials (xⁿ), constants, sin and cos (with linear arguments), exp(ax+b), ln(x), sqrt(ax+b), and their scalar multiples and sums. tan and abs can be evaluated numerically, but are not in its symbolic antiderivative rules. For definite integrals, expressions outside the symbolic rules use Simpson’s rule; an indefinite request shows that a symbolic antiderivative is unavailable. The calculator does not run general substitution, integration-by-parts, or partial-fractions methods.

How accurate is the numerical integration?

Definite integrals use composite Simpson's rule with 1000 subintervals. This gives approximately 10–12 significant digits of accuracy for smooth functions. Results are shown to 8 significant figures.

What does "+ C" mean?

"+C" represents an arbitrary constant of integration. Every antiderivative differs from others by a constant, so the full solution to an indefinite integral is the expression plus any constant. When evaluating a definite integral, the constant cancels out: F(b) + C − (F(a) + C) = F(b) − F(a).