Calculate Value of Indefinite Integral

An indefinite integral represents the antiderivative of a function, which is the reverse process of differentiation. This calculator helps you find the antiderivative of a given function, including polynomial, trigonometric, exponential, and logarithmic functions.

What is an Indefinite Integral?

An indefinite integral, also known as an antiderivative, is a function whose derivative is the original function. Unlike definite integrals, which produce a numerical value, indefinite integrals produce a family of functions that differ by a constant.

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 are used to solve problems involving areas under curves, volumes of solids, and other applications in physics and engineering.

How to Calculate an Indefinite Integral

Calculating an indefinite integral involves finding a function whose derivative matches the original function. Here are the basic rules and techniques:

Basic Rules

Power Rule: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (for n ≠ -1)
Exponential Rule: ∫eˣ dx = eˣ + C
Logarithmic Rule: ∫(1/x) dx = ln|x| + C

Integration Techniques

Substitution Method: Used for composite functions.
Integration by Parts: Used for products of functions.
Partial Fractions: Used for rational functions.

For complex integrals, it may be necessary to use advanced techniques or symbolic computation software.

Common Indefinite Integrals

Here are some common indefinite integrals and their antiderivatives:

Integrand	Antiderivative
∫xⁿ dx	(xⁿ⁺¹)/(n+1) + C (n ≠ -1)
∫eˣ dx	eˣ + C
∫sin(x) dx	-cos(x) + C
∫cos(x) dx	sin(x) + C
∫sec²(x) dx	tan(x) + C
∫1/x dx	ln\|x\| + C

Applications of Indefinite Integrals

Indefinite integrals have numerous applications in various fields:

Physics: Calculating work done by a variable force, determining velocity from acceleration.
Engineering: Finding the displacement from velocity, calculating the area under a curve.
Economics: Determining total cost or revenue from marginal cost or revenue.
Statistics: Calculating probability distributions and cumulative distribution functions.

FAQ

What is the difference between definite and indefinite integrals?: A definite integral produces a numerical value over a specific interval, while an indefinite integral produces a family of functions that differ by a constant.
How do I know when to use the substitution method?: The substitution method is useful when the integrand is a composite function, and you can identify a substitution that simplifies the integral.
What is the constant of integration?: The constant of integration (C) accounts for the infinite number of antiderivatives that differ by a constant. It is necessary when solving indefinite integrals.
Can I integrate any function?: Not all functions have closed-form antiderivatives. Some integrals require advanced techniques or numerical methods.
How do I verify my antiderivative?: You can verify your antiderivative by differentiating it and checking if you get back to the original function.