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What is an indefinite integral?

An indefinite integral, or antiderivative, of f(x) is a family of functions F(x)F(x) whose derivative is f(x)f(x). Unlike a definite integral, there are no bounds—so the result includes an arbitrary constant + C+\,C.

Indefinite integral formula

∫f(x) dx=F(x)+C\int f(x)\,dx = F(x) + C

Here, F′(x)=f(x)F'(x) = f(x) and + C+\,C accounts for any constant whose derivative is zero.

How to solve an indefinite integral step by step

Step 1: Identify the integrand f(x).

Step 2: Recognize a matching antiderivative rule (power rule, substitution, by parts, etc.).

Step 3: Compute F(x)F(x) such that F′(x)=f(x)F'(x)=f(x).

Step 4: Don’t forget the constant of integration + C+\,C.

Example: ∫x2 dx\int x^2\,dx

∫x2 dx=x33+C \int x^2\,dx = \tfrac{x^3}{3} + C

Tips & Tricks

  • Always include + C+\,C—it’s essential for the general solution.
  • Use substitution when the integrand contains a function and its derivative.
  • Memorize common antiderivatives (powers, exponentials, trig) for quick recall.

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