Indefinite Integral Calculator. Enter any function of x into the Indefinite Integral Calculator and get the antiderivative (indefinite integral) returned with the constant of integration + C. Supports polynomials, trigonometric, exponential, logarithmic, and rational functions. Choose your integration variable, select a function type preset, and the calculator computes ∫f(x)dx for you. Also try the Limit Calculator.
An indefinite integral (antiderivative) is the reverse operation of differentiation. For a function f(x), the indefinite integral ∫f(x)dx finds a family of functions F(x) + C such that F'(x) = f(x). The constant C represents an arbitrary constant of integration, since the derivative of any constant is zero. See also our Simpson's Rule Calculator.
The constant of integration C appears because differentiation eliminates any constant. For example, both x² + 5 and x² − 3 have derivative 2x, so ∫2x dx = x² + C covers all possible antiderivatives at once. When you have boundary conditions or initial values, C can be determined.
The Power Rule states that ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, provided n ≠ −1. For example, ∫x³ dx = x⁴/4 + C. When n = −1, the integral is ∫(1/x) dx = ln|x| + C.
Common trig integrals include: ∫sin(x) dx = −cos(x) + C, ∫cos(x) dx = sin(x) + C, and ∫tan(x) dx = −ln|cos(x)| + C. With a coefficient b inside the argument (e.g. sin(bx)), divide by b: ∫sin(bx) dx = −cos(bx)/b + C.
The exponential function is its own antiderivative: ∫eˣ dx = eˣ + C. With a coefficient, ∫e^(bx) dx = e^(bx)/b + C. For other bases, ∫aˣ dx = aˣ / ln(a) + C where a > 0, a ≠ 1.
The most common integration techniques include: the Power Rule (for polynomials), u-Substitution (for composite functions), Integration by Parts (∫u dv = uv − ∫v du, for products), Partial Fractions (for rational functions), and Trigonometric Substitution (for expressions involving √(a²−x²) and similar forms).
To verify an antiderivative F(x), simply differentiate it. If F'(x) = f(x), the answer is correct. For example, if you claim ∫3x² dx = x³ + C, check: d/dx(x³ + C) = 3x² ✓. This calculator's verification row shows this check for supported function types.
An indefinite integral ∫f(x)dx produces a general antiderivative F(x) + C with no numerical value. A definite integral ∫[a to b] f(x) dx uses the Fundamental Theorem of Calculus — it equals F(b) − F(a) and gives a specific number representing the net area under the curve between x = a and x = b. You might also find our Romberg Integration Calculator useful.