Antiderivative Calculator

Antiderivative Calculator: The General Indefinite Integral

The Antiderivative Calculator finds F(x) — the general indefinite integral of any standard function f(x). If F′(x) = f(x), then F(x) is an antiderivative of f(x). Because the derivative of any constant is zero, every antiderivative is a family of functions differing by an additive constant C.

The Fundamental Theorem of Calculus

The deep connection between antiderivatives and definite integrals is: ∫[a→b] f(x)dx = F(b) − F(a). Once you know an antiderivative F(x), you compute any definite integral simply by evaluating F at the two endpoints. This is why finding antiderivatives analytically is so powerful — it eliminates the need for numerical approximation.

Common Antiderivative Rules

• Power Rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.
• Exponential: ∫e^x dx = e^x + C.
• Reciprocal: ∫(1/x) dx = ln|x| + C.
• Trig: ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫tan x dx = −ln|cos x| + C.
• Integration by Parts: ∫u dv = uv − ∫v du — for products like x·e^x.

Why + C Matters

The constant of integration C is mathematically essential. Without initial conditions pinning down a specific C value, the antiderivative is non-unique. In physics, finding particle position requires two constants of integration (initial velocity and position). Omitting C is a common error that leads to significant downstream errors in applied problems.