Integral Calculator
Evaluate an indefinite integral (find an antiderivative and add + C) or a definite integral (compute the signed area between a curve and the x-axis) — with a distinct visual for each mode, full step-by-step work, and a numerical fallback for integrals with no elementary antiderivative.
Background
An indefinite integral ∫ f(x) dx asks: "what family of functions has f(x) as its derivative?" Its answer is always a family, written with a + C since any constant can be added without changing the derivative. A definite integral ∫ab f(x) dx asks a different question: "what is the net signed area between the curve and the x-axis from x=a to x=b?" When an antiderivative F(x) is known, the Fundamental Theorem of Calculus connects the two: ∫ab f(x) dx = F(b) − F(a). When no elementary antiderivative exists, this calculator estimates the definite integral numerically instead.
How to use this calculator
- Choose Indefinite to find an antiderivative, or Definite to compute a signed area between two bounds.
- Type your integrand using ^ for powers and standard function names. Implicit multiplication is supported.
- For definite integrals, enter the lower and upper bounds — these can include pi or e.
- Click Calculate to see the result, the matching visual, and the full step-by-step work.
- If no elementary antiderivative is recognized, definite integrals fall back to a numerical estimate (Simpson's Rule) automatically.
How this calculator works
For indefinite integrals, the calculator matches your integrand against a library of common antiderivative rules — power, exponential, trig, log, arctan, and a few integration-by-parts cases — including generalized versions with a linear inner expression, like sin(3x−1) or (2x+5)^4, handled by u-substitution.
Sums and differences are split apart first using linearity, ∫(f ± g) dx = ∫f dx ± ∫g dx, then each piece is matched and integrated separately before recombining.
For definite integrals, if a symbolic antiderivative F(x) is found, the Fundamental Theorem of Calculus gives the exact value: F(b) − F(a).
If no symbolic antiderivative is recognized — which happens for genuinely non-elementary integrands like e^(x²) — the calculator instead estimates the definite integral numerically using Simpson's Rule, a highly accurate method for smooth functions.
A definite integral measures signed area: regions where the curve dips below the x-axis subtract from the total rather than adding to it, so the net result can be positive, negative, or exactly zero even when the curve clearly encloses visible area.
Formulas & Equations Used
Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1 (generalizes to (ax+b)ⁿ via u-substitution)
Log rule: ∫ 1/x dx = ln|x| + C
Exponential rule: ∫ eᵡ dx = eᵡ + C; more generally ∫ bᵡ dx = bᵡ/ln(b) + C for a constant base b > 0
Trig basics: ∫ sin(x) dx = −cos(x) + C, ∫ cos(x) dx = sin(x) + C, ∫ tan(x) dx = −ln|cos(x)| + C
Inverse trig form: ∫ 1/(1+x²) dx = arctan(x) + C
Integration by parts: ∫ u dv = uv − ∫ v du
Fundamental Theorem of Calculus: ∫ab f(x) dx = F(b) − F(a)
Example Problems & Step-by-Step Solutions
Example 1 — Basic power rule
Evaluate ∫ x³ dx.
Step: Add 1 to the exponent (3+1=4), then divide by the new exponent.
Result: ∫ x³ dx = x⁴/4 + C.
Example 2 — u-substitution with a shift
Evaluate ∫ cos(2x+1) dx.
Step: Let u = 2x+1, so du = 2 dx. ∫ cos(u) du/2 = sin(u)/2.
Result: ∫ cos(2x+1) dx = sin(2x+1)/2 + C.
Example 3 — Integration by parts
Evaluate ∫ x·e^x dx.
Step: Let u=x, dv=e^x dx, so du=dx, v=e^x. ∫ xe^x dx = xe^x − ∫ e^x dx.
Result: ∫ x·e^x dx = e^x(x−1) + C.
Example 4 — Definite integral, positive area
Evaluate ∫03 x² dx.
Step: F(x) = x³/3. F(3) − F(0) = 9 − 0.
Result: ∫03 x² dx = 9.
Example 5 — A full period cancels to zero
Evaluate ∫02π sin(x) dx.
Step: F(x) = −cos(x). F(2π) − F(0) = −1 − (−1) = 0.
Result: 0 — the positive and negative humps of sine over a full period exactly cancel.
Example 6 — No elementary antiderivative
Evaluate ∫01 e^(x²) dx.
Step: No combination of elementary functions differentiates to e^(x²), so the calculator falls back to Simpson's Rule.
Result: ∫01 e^(x²) dx ≈ 1.4627 (numerical estimate).
Frequently Asked Questions
What's the difference between an indefinite and definite integral?
An indefinite integral gives a family of antiderivatives and includes + C. A definite integral gives a single number: the signed area over a specific interval [a, b].
Why does the indefinite mode show three curves instead of one?
Because + C means infinitely many antiderivatives exist, all differing by a vertical shift. Plotting a few of them (like C = −2, 0, 2) makes that "family of curves" idea visible instead of abstract.
Why can a definite integral be negative or zero even though the curve clearly has area?
Because a definite integral measures signed area, not the area you'd get with a ruler and scissors. Any part of the curve below the x-axis subtracts from the total, which is exactly why ∫sin(x) dx over a full period comes out to 0.
Does this calculator solve every possible integral?
No calculator can — some integrands, like e^(x²), simply have no elementary antiderivative. This calculator recognizes a broad set of common patterns (power, exponential, trig, log, arctan, and several integration-by-parts cases) and automatically switches to a numerical estimate for definite integrals when no symbolic match is found.
What does "u-substitution" mean in the rule shown?
It means the integrand has a linear inner expression (like 3x−1 inside a sine or exponent), and substituting u for that inner expression turns the integral into a basic form, which then gets divided by the inner expression's slope after integrating.
Why does the calculator warn me about the domain?
Some functions aren't defined everywhere — square roots of negative numbers, logarithms of non-positive numbers, and division by zero all break a formula at specific inputs. The calculator flags this instead of silently producing a wrong number.
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