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. Expressions that require substitution, integration by parts, or partial fractions (beyond ln) fall back to high-precision numerical integration.

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).

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

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

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

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

Exponents: x^3 or x^(1/2)
Multiplication: 3*x^2
Functions: sin(x), cos(x), exp(x), ln(x), sqrt(x)
Combinations: 3*x^2 + sin(2*x) - exp(-x)