integrate 1 dx
Understand the Problem
The question is asking us to find the integral of the function 1 with respect to x. This is a fundamental calculus problem which involves determining the antiderivative of the constant function 1, resulting in a linear function plus a constant of integration.
Answer
The integral of 1 with respect to x is $x + C$.
Answer for screen readers
The integral of the function 1 with respect to x is $x + C$.
Steps to Solve
- Identify the function to integrate
The function we need to integrate is the constant function $1$.
- Set up the integral
To find the integral of $1$ with respect to $x$, we write it as: $$ \int 1 , dx $$
- Calculate the integral
The integral of a constant $c$ with respect to $x$ is given by: $$ \int c , dx = cx + C $$ where $C$ is the constant of integration. Here, $c = 1$, so we have: $$ \int 1 , dx = 1 \cdot x + C = x + C $$
- State the result
The final result of the integral is: $$ x + C $$
The integral of the function 1 with respect to x is $x + C$.
More Information
The integral we calculated, $x + C$, represents a family of linear functions, where $C$ can take any real value. This concept is fundamental in calculus as it allows us to find functions whose derivatives equal specific values.
Tips
- A common mistake is forgetting to include the constant of integration $C$ when calculating the integral. Always remember that indefinite integrals have an arbitrary constant.
