Free Online Integral Calculator (Definite & Indefinite)

The Difference Between Definite and Indefinite Integration

Integration is a fundamental concept in calculus, representing the inverse process of differentiation. Your calculator handles both primary forms:

• Indefinite Integral (Antiderivative): This is the reverse process of finding the derivative. It results in a family of functions, always denoted by the addition of the constant of integration ($+C$). Our tool shows you the function that, when differentiated, would return the original function you entered.

• Definite Integral: This represents the area under a curve between two specified points, known as the upper ($b$) and lower ($a$) bounds. The result is always a single numerical value.

By using our Advanced Integral Calculator, you can quickly verify your manual work or solve complex expressions instantly.

Key Rules and Formulas for Integration

To better understand the solution your calculator provides, here are the most important rules governing integration:

• The Power Rule for Integration: If you integrate $x^n$, the result is $\frac{x^{n+1}}{n+1} + C$. (As long as $n \neq -1$).

• The Constant Rule: The integral of a constant $k$ is $kx + C$.

• The Constant Multiple Rule: $\int k \cdot f(x) dx = k \int f(x) dx$

• The Sum/Difference Rule: You can integrate terms in a polynomial separately.

Related Calculus Tools and Resources

Integration and differentiation are inseparable. If you need help with the opposite operation, try our related utility:

➡️ Derivative Calculator – Find the rate of change and the slope of a function instantly.

The definite integral is mathematically defined as the limit of a Riemann Sum. This crucial concept relies entirely on understanding how functions behave near a given point. Solve the fundamental building blocks of calculus by using our Limit Calculator before attempting advanced integration techniques.