What is the integral of 10x?
Understand the Problem
The question is asking for the integral of the function 10x with respect to x. The integral represents the area under the curve of the function on a graph and can be computed using the power rule for integration.
Answer
The integral of $10x$ with respect to $x$ is $5x^2 + C$.
Answer for screen readers
The integral of the function $10x$ with respect to $x$ is $5x^2 + C$.
Steps to Solve
- Identify the function to integrate
The function we want to integrate is $10x$.
- Apply the power rule for integration
The power rule states that the integral of $x^n$ is given by:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$
For the function $10x$, we can rewrite it as $10x^1$, so we will apply the power rule where $n = 1$.
- Compute the integral
Using the power rule:
$$ \int 10x , dx = 10 \cdot \frac{x^{1+1}}{1+1} + C = 10 \cdot \frac{x^2}{2} + C $$
- Simplify the result
Now simplify the expression:
$$ 10 \cdot \frac{x^2}{2} + C = 5x^2 + C $$
Thus, the integral of $10x$ with respect to $x$ is $5x^2 + C$.
The integral of the function $10x$ with respect to $x$ is $5x^2 + C$.
More Information
Integrals provide the area under curves, and when you integrate a linear function like $10x$, you obtain a quadratic function. The constant $C$ represents the constant of integration which accounts for any vertical shift in the function.
Tips
- Forgetting to include the constant of integration $C$ is a common mistake. Always include it in indefinite integrals.
