What is the integral of 2x dx?
Understand the Problem
The question is asking for the integral of the function 2x with respect to x. To solve this, we will apply the basic rules of integration, specifically looking for an antiderivative of the polynomial function.
Answer
The integral of $2x$ with respect to $x$ is $x^2 + C$.
Answer for screen readers
The integral of the function $2x$ with respect to $x$ is:
$$ x^2 + C $$
Steps to Solve
- Identify the function to integrate
We need to find the integral of the function $2x$ with respect to $x$.
- Apply the Power Rule of Integration
The Power Rule states that for a function of the form $x^n$, the integral is given by:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$
where $C$ is the constant of integration. In our case, we will treat $2x$ as $2 \cdot x^1$.
- Integrate the function using the Power Rule
Using the Power Rule, we first apply the rule to $x^1$:
$$ \int 2x , dx = 2 \int x^1 , dx = 2 \cdot \frac{x^{1+1}}{1+1} + C $$
This simplifies to:
$$ = 2 \cdot \frac{x^2}{2} + C $$
- Simplify the expression
Now, we simplify our result:
$$ = x^2 + C $$
Thus, the integral of $2x$ is $x^2 + C$.
The integral of the function $2x$ with respect to $x$ is:
$$ x^2 + C $$
More Information
The integral we found represents the area under the curve of the function $2x$ over a particular interval, with $C$ being a constant that reflects the family of antiderivatives. Integrals are fundamental in calculus and are used in various applications, such as calculating areas and solving differential equations.
Tips
- Confusing the Power Rule with the derivative, leading to incorrect calculations.
- Forgetting to include the constant of integration $C$ when writing the antiderivative.
