What is the integral of dx?

Understand the Problem

The question is asking for the indefinite integral of the function 1 with respect to x, which is a foundational concept in calculus.

Answer

$$ x + C $$
Answer for screen readers

The indefinite integral of the function 1 with respect to $x$ is:

$$ x + C $$

Steps to Solve

  1. Identify the integral to be solved

We need to find the indefinite integral of the function 1 with respect to $x$. This can be written as:

$$ \int 1 , dx $$

  1. Apply the integral rule

The integral of a constant function like 1 is straightforward. The rule states that:

$$ \int k , dx = kx + C $$

where $k$ is a constant, and $C$ is the constant of integration.

  1. Substitute the constant

Since $k$ is 1 in our case, we substitute $1$ for $k$:

$$ \int 1 , dx = 1 \cdot x + C $$

  1. Simplify the expression

This simplifies to:

$$ x + C $$

The indefinite integral of the function 1 with respect to $x$ is:

$$ x + C $$

More Information

The result $x + C$ represents a family of linear functions where $C$ is any real number. This shows that the slope of the function is constant, and the function is a straight line at a vertical distance of $C$ from the x-axis.

Tips

  • Forgetting to include the constant of integration $C$. Every indefinite integral must include this constant since the process of integration can yield an infinite number of functions differing by a constant value.
  • Misunderstanding the integral symbol. Some might confuse indefinite with definite integrals, where limits are specified, resulting in a specific numerical answer without the constant of integration.

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