Symbolab Indefinite Integral Calculator

This Symbolab indefinite integral calculator helps you find the antiderivative of functions with step-by-step solutions. Whether you're a student studying calculus or a professional applying mathematical principles, this tool provides accurate results and explanations.

What is an Indefinite Integral?

An indefinite integral represents the antiderivative of a function. Unlike definite integrals, which calculate the area under a curve between two points, indefinite integrals find all possible functions whose derivative is the original function. The result is expressed with a constant of integration, denoted by the symbol "C".

The general form of an indefinite integral is:

∫f(x) dx = F(x) + C

where F(x) is the antiderivative of f(x), and C is the constant of integration.

Indefinite integrals are fundamental in calculus and have applications in physics, engineering, and economics. They help solve differential equations, find areas under curves, and determine the behavior of functions over intervals.

How to Use This Calculator

Enter the function you want to integrate in the input field. For example, type "x^2 + 3x" for the function f(x) = x² + 3x.
Select the variable of integration (usually "x").
Click the "Calculate" button to compute the indefinite integral.
Review the result, which includes the antiderivative and the constant of integration.
Use the optional graph to visualize the function and its antiderivative.

Tip: For complex functions, use proper mathematical notation. For example, write "sin(x)" instead of "sinx" and "e^x" instead of "exp(x)".

Formula Used

The calculator uses standard rules of integration to find the antiderivative of a function. Some common integration rules include:

Power Rule: ∫x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1)
Exponential Rule: ∫e^x dx = e^x + C
Trigonometric Rules:
- ∫sin(x) dx = -cos(x) + C
- ∫cos(x) dx = sin(x) + C
- ∫sec²(x) dx = tan(x) + C
Logarithmic Rule: ∫(1/x) dx = ln|x| + C

The calculator combines these rules and others to handle more complex functions. For functions that cannot be integrated using elementary functions, the calculator may return an error or an expression involving special functions.

Worked Examples

Example 1: Basic Polynomial

Find the indefinite integral of f(x) = 3x² + 2x.

∫(3x² + 2x) dx = x³ + x² + C

Example 2: Trigonometric Function

Find the indefinite integral of f(x) = cos(x).

∫cos(x) dx = sin(x) + C

Example 3: Exponential Function

Find the indefinite integral of f(x) = e^(2x).

∫e^(2x) dx = (1/2)e^(2x) + C

Frequently Asked Questions

What is the difference between definite and indefinite integrals?: A definite integral calculates the area under a curve between two points, while an indefinite integral finds all possible antiderivatives of a function, expressed with a constant of integration.
How do I know if a function can be integrated?: Most elementary functions can be integrated using standard rules. For more complex functions, the calculator may return an expression involving special functions or an error if integration is not possible with elementary functions.
What is the constant of integration (C) for?: The constant of integration (C) represents the family of all possible antiderivatives for a given function. It accounts for the infinite number of functions that could have the same derivative.
Can this calculator handle piecewise functions?: This calculator is designed for continuous functions. Piecewise functions may require manual integration or specialized tools.
How accurate are the results from this calculator?: The calculator uses standard integration rules and provides accurate results for elementary functions. For complex functions, the results may involve special functions or approximations.