Find The General Indefinite Integral Calculator

The general indefinite integral calculator finds the antiderivative of a function. This tool helps solve calculus problems by computing integrals of common functions, including polynomials, trigonometric functions, exponential functions, and more.

What is an Indefinite Integral?

An indefinite integral represents the antiderivative of a function. Unlike definite integrals, which compute the area under a curve between two points, indefinite integrals find all possible antiderivatives of a function, including a constant of integration (C).

The general form of an indefinite integral is written as:

Indefinite Integral Formula

∫f(x) dx = F(x) + C

Where:

f(x) is the integrand (function to be integrated)
F(x) is the antiderivative of f(x)
C is the constant of integration

Indefinite integrals are fundamental in calculus for solving differential equations, finding areas under curves, and analyzing functions.

How to Find the Indefinite Integral

Finding the indefinite integral involves identifying the antiderivative of a function. Here are the basic steps:

Identify the type of function (polynomial, trigonometric, exponential, etc.).
Apply the appropriate integration rule.
Add the constant of integration (C) to the result.

For complex functions, you may need to use integration techniques such as substitution, integration by parts, or partial fractions.

Common Integral Formulas

Here are some common indefinite integral formulas:

Basic Power Rule

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (for n ≠ -1)

Exponential Function

∫eˣ dx = eˣ + C

Natural Logarithm

∫(1/x) dx = ln|x| + C

Trigonometric Functions

∫sin(x) dx = -cos(x) + C

∫cos(x) dx = sin(x) + C

∫sec²(x) dx = tan(x) + C

Example Calculations

Let's look at some example calculations:

Example 1: Polynomial Function

Find ∫(3x² + 2x) dx

Solution:

∫(3x² + 2x) dx = x³ + x² + C

Example 2: Trigonometric Function

Find ∫sin(x) dx

Solution:

∫sin(x) dx = -cos(x) + C

Example 3: Exponential Function

Find ∫eˣ dx

Solution:

∫eˣ dx = eˣ + C

FAQ

What is the difference between definite and indefinite integrals?

Definite integrals compute the area under a curve between two points and yield a numerical value. Indefinite integrals find all possible antiderivatives of a function and include a constant of integration.

Why is the constant of integration (C) important?

The constant of integration (C) represents the family of curves that have the same derivative. It accounts for the infinite number of possible antiderivatives that differ by a constant.

Can all functions be integrated?

Not all functions have closed-form antiderivatives. Some functions require advanced techniques or numerical methods to approximate their integrals.