What is the integral of 1 dx?
Understand the Problem
The question is asking for the indefinite integral of the function 1 with respect to x. The integral of a constant function can be calculated using the formula for integration, which results in the product of the constant and the variable of integration plus a constant of integration.
Answer
$$ x + C $$
Answer for screen readers
The indefinite integral of the function 1 with respect to $x$ is:
$$ x + C $$
Steps to Solve
- Identify the integral to calculate
We want to find the indefinite integral of the constant function 1 with respect to $x$:
$$ \int 1 , dx $$
- Apply the integration rule for constants
The rule for integrating a constant $c$ with respect to $x$ is:
$$ \int c , dx = cx + C $$
where $C$ is the constant of integration. So, for our case with $c = 1$:
$$ \int 1 , dx = 1 \cdot x + C $$
- Simplify the expression
The expression can be simplified to:
$$ x + C $$
Thus, we have found the integral of the constant function 1 with respect to $x$.
The indefinite integral of the function 1 with respect to $x$ is:
$$ x + C $$
More Information
The constant of integration $C$ represents an infinite number of possible vertical shifts of the function. This is because there are many functions that, when derived, yield 1, such as $x + 1$, $x + 2$, etc. The integral captures all these functions.
Tips
- Forgetting to add the constant of integration $C$ after finding the indefinite integral. Always remember to include it.
- Misidentifying the integral type, particularly with respect to constant functions. This integral should not be confused with integrals of variable functions.
