Indefinite Integrals Calculator

Indefinite integrals are fundamental in calculus, representing the antiderivative of a function. This calculator helps you compute indefinite integrals for various functions, providing both the result and the step-by-step process.

What is Indefinite Integration?

Indefinite integration is the process of finding the antiderivative of a function. Unlike definite integration, which calculates the area under a curve between two points, indefinite integration finds a general function whose derivative is the original function.

The result of an indefinite integral is expressed with a constant of integration (usually denoted as C), representing the infinite number of possible antiderivatives that differ by a constant.

General Form: ∫f(x) dx = F(x) + C

Where:

∫ is the integral symbol
f(x) is the integrand (the function to be integrated)
dx indicates integration with respect to x
F(x) is the antiderivative of f(x)
C is the constant of integration

Basic Integration Rules

Here are some fundamental rules for integrating common functions:

Power Rule

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

Example: ∫x² dx = (x³)/3 + C

Constant Multiple Rule

∫k·f(x) dx = k·∫f(x) dx

Where k is a constant.

Sum/Difference Rule

∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx

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

∫csc(x)cot(x) dx = -csc(x) + C

∫sec(x)tan(x) dx = sec(x) + C

How to Use This Calculator

Enter the function you want to integrate in the input field.
Select the variable of integration (usually x).
Click "Calculate" to compute the indefinite integral.
Review the result, which includes the antiderivative and the constant of integration.
Use the chart to visualize the function and its antiderivative.

Note: This calculator handles basic algebraic, exponential, logarithmic, and trigonometric functions. For more complex functions, consult calculus textbooks or advanced symbolic computation tools.

Common Integration Examples

Example 1: Polynomial Function

Find ∫(3x² + 2x - 5) dx

∫(3x² + 2x - 5) dx = x³ + x² - 5x + C

Example 2: Exponential Function

Find ∫eˣ dx

∫eˣ dx = eˣ + C

Example 3: Trigonometric Function

Find ∫sin(x) dx

∫sin(x) dx = -cos(x) + C

Example 4: Rational Function

Find ∫(1/x) dx

∫(1/x) dx = ln|x| + C

FAQ

What is the difference between definite and indefinite integrals?

Definite integrals calculate the exact area under a curve between two points, while indefinite integrals find the general antiderivative of a function, including a constant of integration.

Why is the constant of integration (C) necessary?

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

What functions can this calculator integrate?

This calculator handles basic algebraic, exponential, logarithmic, and trigonometric functions. For more complex functions, advanced symbolic computation tools are recommended.

How accurate are the results from this calculator?

The calculator provides accurate results for the functions it supports, following standard calculus rules. For complex or specialized functions, manual verification is advised.