What is the antiderivative of 1?
Understand the Problem
The question is asking for the antiderivative (or indefinite integral) of the constant function 1. This involves determining the function whose derivative is 1.
Answer
The antiderivative of 1 is $x + C$.
Answer for screen readers
The antiderivative of the constant function 1 is $x + C$, where $C$ is the constant of integration.
Steps to Solve
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Identify the constant function The function we are working with is the constant function $f(x) = 1$.
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Use the power rule for integration For any constant function $c$, the antiderivative can be calculated using the formula:
$$ \int c , dx = cx + C $$
where $C$ is the constant of integration. -
Apply the formula for the antiderivative In this case, since our constant $c$ is 1, we plug it into the formula:
$$ \int 1 , dx = 1 \cdot x + C $$ -
Write the final result We simplify this to the final result:
$$ x + C $$
The antiderivative of the constant function 1 is $x + C$, where $C$ is the constant of integration.
More Information
The antiderivative represents a family of functions that all have the same derivative, which is 1. The constant of integration $C$ accounts for any vertical shift in the function since differentiating any constant results in zero.
Tips
- Forgetting to include the constant of integration $C$ is a common mistake. Always remember that antiderivatives represent families of functions.
