Our Integral Calculator handles both indefinite integrals (antiderivatives) and definite integrals (area under a curve). Enter an expression in terms of x and get the antiderivative instantly with all steps explained. For definite integrals, provide the lower and upper bounds to compute the exact numerical result. Supports polynomials, trigonometric functions (sin, cos, tan, sec, csc, cot), exponential (e^x), logarithms (ln x), square roots, and common composite functions. When symbolic integration is not possible, the calculator falls back to high-precision numerical integration using Simpson's rule.
Use ^ for powers, * for multiply, sin/cos/exp/ln/sqrt for functions
Quick Examples
Type your function using x as the variable. Use ^ for powers, * for multiplication, and standard function names like sin, cos, exp, ln, sqrt.
Select Indefinite for the general antiderivative (with +C), or Definite to compute the area under the curve between two bounds.
For definite integrals, enter the lower bound a and upper bound b. The result will be the net signed area F(b) - F(a).
See which integration rules were applied (power rule, substitution, by parts) with each step explained clearly.
Integration is the reverse of differentiation. The indefinite integral ∫f(x)dx gives a family of functions F(x)+C. The definite integral ∫[a,b]f(x)dx gives the net signed area under the curve from x=a to x=b, computed using the Fundamental Theorem of Calculus: F(b) - F(a).
∫f(x)dx = F(x) + C | ∫[a to b]f(x)dx = F(b) - F(a)
Use ^ for powers: x^3 means x³. Use * for multiplication: 2*x or just 2x
Power rule: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (for n ≠ -1). Special case: ∫x⁻¹dx = ln|x| + C
Trig integrals: ∫sin(x)dx = -cos(x)+C, ∫cos(x)dx = sin(x)+C, ∫sec²(x)dx = tan(x)+C
Exponential: ∫eˣdx = eˣ+C, ∫aˣdx = aˣ/ln(a)+C
Logarithm: ∫ln(x)dx = x·ln(x) - x + C (use integration by parts)
For ∫f(ax+b)dx, substitute u=ax+b: result is F(ax+b)/a + C
Integration is one of the two fundamental operations in calculus, alongside differentiation. While derivatives measure instantaneous rates of change, integrals accumulate quantities — computing areas under curves, volumes of solids, total distance from velocity, and accumulated quantities from rates. An indefinite integral (antiderivative) reverses differentiation: the integral of 2x is x² + C, where C is the constant of integration. A definite integral evaluates the net signed area between a function and the x-axis over a specific interval. The Fundamental Theorem of Calculus connects the two, stating that the definite integral from a to b equals F(b) minus F(a), where F is any antiderivative of the integrand. Integration techniques range from straightforward power rule applications to sophisticated methods like substitution, integration by parts, partial fractions, and trigonometric identities. Some functions — like e^(-x²), central to the normal distribution in statistics — have no closed-form antiderivative at all, requiring numerical methods such as Simpson's rule or Gaussian quadrature. Whether you are a student working through a calculus course, an engineer computing work or flux, or a physicist evaluating probability amplitudes, a solid grasp of integration is indispensable.
An integral calculator computes the antiderivative (indefinite integral) or definite integral of a mathematical function.
For indefinite integrals, it finds F(x)+C where F'(x)=f(x). For definite integrals, it evaluates the net signed area under the curve f(x) between two bounds using the Fundamental Theorem of Calculus.
This calculator supports:
Use * for multiplication, ^ for exponentiation, and standard parentheses for grouping.
Integrating a function means finding a new function whose derivative equals the original — the reverse of differentiation.
The result, called the antiderivative or indefinite integral, is written F(x) + C, where the derivative F'(x) equals the integrand f(x) and C is an arbitrary constant. Geometrically, a definite integral measures the net signed area between the curve and the x-axis over an interval.
According to Wolfram MathWorld, an integral is a mathematical object that can represent area, accumulated change, or a generalized sum. This dual meaning — antidifferentiation and area — is why integration underpins so much of calculus and applied mathematics.
To find an integral, identify the function's structure and match it to a known rule.
First try the power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1. For sums, integrate each term separately; for constant multiples, pull the constant outside. When the integrand is a composition times an inner derivative, use u-substitution; when it is a product, try integration by parts.
For a definite integral, find the antiderivative F(x), then compute F(b) − F(a). For example, ∫₀¹ x² dx = [x³/3]₀¹ = 1/3 − 0 = 1/3. Khan Academy walks through each of these techniques with practice problems.
The core integration formulas are:
Standard trigonometric results include ∫sin(x) dx = −cos(x) + C and ∫cos(x) dx = sin(x) + C.
Two structural methods extend these: u-substitution reverses the chain rule, and integration by parts, ∫u dv = uv − ∫v du, reverses the product rule. The NIST Digital Library of Mathematical Functions (DLMF) catalogs these elementary results alongside integrals of special functions, making it a reliable reference for verifying any formula.
An indefinite integral produces a function: ∫f(x) dx = F(x) + C, a whole family of antiderivatives differing by the constant C.
A definite integral produces a single number: ∫[a,b] f(x) dx equals the net signed area between the curve and the x-axis from x = a to x = b. The Fundamental Theorem of Calculus links them, stating that a definite integral equals F(b) − F(a) for any antiderivative F.
For instance, ∫₀^π sin(x) dx = [−cos(x)]₀^π = −cos(π) − (−cos(0)) = 1 + 1 = 2. Encyclopaedia Britannica describes this theorem as the central result unifying the two branches of calculus.
u-substitution simplifies an integral when the integrand contains a composite function multiplied by the derivative of its inner part.
Set u equal to the inner expression, compute du = u'(x) dx, and rewrite the integral entirely in terms of u. For example, in ∫2x·cos(x²) dx, let u = x² so du = 2x dx; the integral becomes ∫cos(u) du = sin(u) + C = sin(x²) + C.
For definite integrals, either convert the limits to u-values or back-substitute before evaluating. Khan Academy notes that recognizing the inner-function-and-its-derivative pattern is the key skill for applying substitution confidently.
Integration accumulates quantities, making it essential across science and engineering:
Wolfram MathWorld highlights that integrals also compute arc lengths, surface areas, and volumes of revolution. This calculator supports these applications by returning both symbolic antiderivatives and precise numerical values.
Some functions, such as e^(−x²) and sin(x)/x, have no closed-form antiderivative expressible with elementary functions.
In these cases numerical integration approximates the definite integral by sampling the function. Simpson's Rule fits parabolas across pairs of subintervals: ∫[a,b] f(x) dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + f(xₙ)], where h = (b − a)/n and n is even.
Its error scales as O(h⁴), far better than the Trapezoidal Rule's O(h²). The NIST DLMF documents these quadrature methods, including Gaussian quadrature, which achieves high accuracy with few sample points.
Consider ∫₁^e (1/x) dx. Because the integrand is 1/x, the antiderivative is ln|x|, not a power-rule result.
Apply the Fundamental Theorem of Calculus: ∫₁^e (1/x) dx = [ln|x|]₁^e = ln(e) − ln(1) = 1 − 0 = 1. The exact value is therefore 1.
As a second example, ∫₀¹ eˣ dx = [eˣ]₀¹ = e¹ − e⁰ = e − 1 ≈ 1.71828. These worked steps mirror what the calculator displays: identify the antiderivative, substitute the upper and lower bounds, and subtract. Verifying results against Wolfram MathWorld's integral tables is a good habit for confirming both the formula and the arithmetic.
Britannica emphasizes checking every antiderivative by differentiating it back to the original integrand.
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