Indefinite integral

The indefinite integral describes the inverse process of differentiation. While differential calculus measures how a function changes, integral calculus tells which function produces a given rate of change. The result is called an antiderivative.

 

 

Antiderivative and notation

If a function \( \large F(x) \) has the derivative \( \large F'(x)=f(x) \), then \( \large F \) is an antiderivative of \( \large f \). It is written as:

 

$$ \large \int f(x)\,dx \;=\; F(x) + C $$

 

Here, \( \large C \) is the constant of integration, representing that many different functions can have the same derivative. The constant does not affect the slope but shifts the graph vertically.

 

 

Example 1: Basic power rule

Find the antiderivative of \( \large f(x) = x^n \), where \( \large n \neq -1 \). We look for a function \( \large F \) whose derivative gives \( \large x^n \). This gives:

 

$$ \large \int x^n\,dx \;=\; \frac{x^{n+1}}{n+1} + C $$

 

By adding 1 to the exponent and dividing by the new exponent, we obtain the general antiderivative for powers of \( \large x \).

 

 

Example 2: Sum and constant factor

If the function consists of several terms, each term can be integrated separately. The following rules apply:

 

$$ \large \int \big(f(x) + g(x)\big)\,dx \;=\; \int f(x)\,dx + \int g(x)\,dx $$

$$ \large \int k \cdot f(x)\,dx \;=\; k \cdot \int f(x)\,dx $$

 

These rules make it possible to find antiderivatives for composite expressions by working term by term.

 

 

Example 3: Applied calculation

Find the antiderivative of \( \large f(x) = 3x^2 - 4x + 1 \):

 

$$ \large \int (3x^2 - 4x + 1)\,dx \;=\; x^3 - 2x^2 + x + C $$

 

The resulting function \( \large F(x) = x^3 - 2x^2 + x + C \) has the derivative \( \large F'(x)=3x^2 - 4x + 1 \).

 

 

Verification by differentiation

A good way to check the result is to differentiate the antiderivative again. If we obtain the original function \( \large f(x) \), the calculation is correct. This also shows how integration and differentiation are inverse operations.

 

 

The meaning of the integration constant

The family of indefinite integrals consists of infinitely many functions that differ only by a constant. The constant \( \large C \) has a geometric meaning as a vertical shift: all antiderivatives have the same shape but lie at different heights on the coordinate plane.

 

 

Summary

The indefinite integral is used to find antiderivatives, that is, functions whose derivatives correspond to a given \( \large f(x) \). Integration thus “reverses” differentiation. Each antiderivative differs only by a constant, and correctness can always be checked by differentiating the result.