Integral Calculator — Step-by-Step Solutions

The Integral Calculator computes indefinite integrals (antiderivatives) of mathematical functions. It handles polynomials, trigonometric functions, exponentials, and many other common function types.

Enter your function and integration variable to find the antiderivative. Results are displayed in clear mathematical notation with the constant of integration included.

Expression f(x)

Variable

Integral

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Your calculations will appear here

How to Use the Integral Calculator

Enter the function you want to integrate (e.g. x^2 + 3*x).
Specify the variable of integration (default: x).
The calculator attempts symbolic integration and displays the antiderivative with the constant of integration C.

Formula and Theory

Integration is the reverse of differentiation. The indefinite integral (antiderivative) of a function f(x) is a function F(x) such that F'(x) = f(x). The result always includes an arbitrary constant C because the derivative of any constant is zero.

The Fundamental Theorem of Calculus connects differentiation and integration: if F is an antiderivative of f, then the definite integral from a to b of f(x)dx = F(b) - F(a).

Worked Examples

Example 1: Power rule

Problem: Find the integral of x^3

Solution: Using the power rule: integral of x^n = x^(n+1)/(n+1) + C

Answer: x^4/4 + C

Example 2: Trigonometric integral

Problem: Find the integral of cos(x)

Solution: The antiderivative of cos(x) is sin(x)

Answer: sin(x) + C

Example 3: Polynomial integral

Problem: Find the integral of 3x^2 + 2x - 5

Solution: Integrate term by term: 3*x^3/3 + 2*x^2/2 - 5x

Answer: x^3 + x^2 - 5x + C

Frequently Asked Questions

What is the constant of integration?

The constant C appears because many functions have the same derivative. For example, x^2, x^2 + 5, and x^2 - 3 all have derivative 2x. The constant captures this ambiguity.

What is the difference between indefinite and definite integrals?

An indefinite integral gives a family of functions (antiderivatives) plus C. A definite integral evaluates between two bounds and gives a specific numerical value (the area under the curve).

Can all functions be integrated?

Not all functions have elementary antiderivatives. Famous examples include e^(x^2) and sin(x)/x. These can be evaluated numerically but not expressed as simple formulas.

What are common integration techniques?

Key techniques include: substitution (reverse chain rule), integration by parts (reverse product rule), partial fractions, and trigonometric substitution.