Calculator That Performs Indefinite Integrals

This calculator performs indefinite integrals of functions with respect to a variable. It provides exact solutions when possible and shows the integration process step-by-step.

What is an indefinite integral?

An indefinite integral represents the antiderivative of a function. It finds the family of functions whose derivative is the original function. The result is expressed with a constant of integration, denoted as C.

Indefinite Integral Formula

∫f(x) dx = F(x) + C

Where:

f(x) is the integrand (function to integrate)
dx indicates integration with respect to x
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. They provide a general solution rather than a specific value.

How to calculate indefinite integrals

Calculating indefinite integrals involves finding the antiderivative of a function. Here are the basic rules:

Basic Integration Rules

∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (for n ≠ -1)
∫eˣ dx = eˣ + C
∫aˣ dx = (aˣ)/ln(a) + C (for a > 0, a ≠ 1)
∫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

Integration Techniques

Substitution (change of variables)
Integration by parts
Partial fractions
Trigonometric identities
Special functions (Bessel, Legendre, etc.)

Note

Some functions do not have elementary antiderivatives. In such cases, special functions or numerical methods may be required.

Examples of indefinite integrals

Here are some common examples of indefinite integrals:

Example 1: Polynomial Function

Find ∫(3x² + 2x + 1) dx

Solution:

∫(3x² + 2x + 1) dx = 3∫x² dx + 2∫x dx + ∫1 dx = x³ + x² + x + C

Example 2: Exponential Function

Find ∫eˣ dx

Solution:

∫eˣ dx = eˣ + C

Example 3: Trigonometric Function

Find ∫sin(x) dx

Solution:

∫sin(x) dx = -cos(x) + C

Example 4: Rational Function

Find ∫(1/x) dx

Solution:

∫(1/x) dx = ln|x| + C

FAQ

What is the difference between definite and indefinite integrals?: An indefinite integral finds the family of functions whose derivative is the original function, while a definite integral calculates the exact area under a curve between specified limits.
When is the constant of integration needed?: The constant of integration (C) is needed because indefinite integrals represent a family of functions that differ by a constant. It's only necessary when solving differential equations or finding particular solutions.
Can all functions be integrated?: No, some functions do not have elementary antiderivatives. In such cases, special functions or numerical methods may be required.
How do I know if my integral is correct?: You can verify your integral by differentiating the result. If you get back the original function, your integral is correct.
What are some common applications of indefinite integrals?: Indefinite integrals are used in solving differential equations, finding areas under curves, analyzing functions, and in physics and engineering problems.