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: polynomials (x^2, 3*x^4), trigonometric functions (sin(x), cos(x), tan(x), sec(x), csc(x), cot(x)), exponentials (exp(x) or e^x), natural logarithm (ln(x)), square root (sqrt(x)), and composite functions. Use * for multiplication, ^ for exponentiation, and standard parentheses for grouping.
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