integral of 2xdx
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 power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n is not equal to -1.
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 given by:
$$ x^2 + C $$
Steps to Solve
- Identify the function to integrate
We are looking to integrate the function $2x$.
- Apply the power rule for integration
Since the function can be expressed as $2x = 2x^1$, we can apply the power rule for integration. According to the power rule:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$
Here, ( n = 1 ). Therefore, we can rewrite the integral as:
$$ \int 2x^1 , dx $$
- Calculate the integral using the power rule
Now, apply the power rule:
$$ \int 2x^1 , dx = 2 \cdot \frac{x^{1+1}}{1+1} + C $$
This simplifies to:
$$ 2 \cdot \frac{x^2}{2} + C $$
- Simplify the expression
The $2$ in the numerator and denominator cancel out:
$$ x^2 + C $$
Thus, the final result of the integration is
$$ x^2 + C $$
The integral of the function $2x$ with respect to $x$ is given by:
$$ x^2 + C $$
More Information
The constant $C$ represents the constant of integration, which comes from the fact that the integral can have infinitely many functions differing by a constant. This result indicates that if you differentiate $x^2 + C$, you will get back the original function $2x$.
Tips
- Forgetting to include the constant of integration $C$ after the integral.
- Misapplying the power rule, for example by not correctly increasing the exponent by 1.
