integral of 10x
Understand the Problem
The question is asking for the integral of the function 10x. This involves finding the antiderivative of the expression with respect to x.
Answer
The integral of $10x$ is $5x^2 + C$.
Answer for screen readers
The integral of the function $10x$ is $5x^2 + C$.
Steps to Solve
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Identify the function
The function we need to integrate is $f(x) = 10x$. -
Apply the power rule for integration
The power rule states that the integral of $x^n$ is $\frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration.
Here, the function $f(x)$ can be expressed as $10x^1$. -
Integrate the function
Using the power rule, we find the integral:
$$ \int 10x , dx = 10 \cdot \frac{x^{1+1}}{1+1} + C = 10 \cdot \frac{x^2}{2} + C $$ -
Simplify the expression
Now we simplify the expression:
$$ \int 10x , dx = 5x^2 + C $$
The integral of the function $10x$ is $5x^2 + C$.
More Information
The constant $C$ represents the constant of integration, which accounts for any number that could have been differentiated away. The integral is widely used in calculus to find areas under curves and in various applications, including physics and engineering.
Tips
- Forgetting to add the constant of integration $C$ is a common mistake. Always remember that indefinite integrals include this constant.
- Misapplying the power rule, such as increasing the power or coefficient incorrectly.
