integral of 3x
Understand the Problem
The question is asking for the indefinite integral of the function 3x with respect to x. The integral is a fundamental concept in calculus that represents the accumulation of quantities and is often used to find areas under curves.
Answer
$\frac{3x^2}{2} + C$
Answer for screen readers
The final answer is $\frac{3x^2}{2} + C$
Steps to Solve

Identify the integral to be solved
We are given the function $3x$ and need to find its indefinite integral with respect to $x$.
$$ \int 3x \ dx $$

Apply the power rule of integration
The power rule for integration states that for any function $x^n$, its integral is given by:
$$ \int x^n \ dx = \frac{x^{n+1}}{n+1} + C $$
In this case, $x$ is raised to the power of 1. Therefore, $ x^n = x^1 $.

Multiply by the constant
Since we have a constant 3 multiplying $x$, we can factor out the constant and then apply the integration rule:
$$ \int 3x \ dx = 3 \int x \ dx $$

Integrate $x$ using the power rule
Use the power rule to integrate $x$:
$$ \int x \ dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2} $$

Multiply the result by the constant
Now multiply by the constant 3:
$$ 3 \cdot \frac{x^2}{2} = \frac{3x^2}{2} $$

Add the constant of integration $C$
Finally, include the constant of integration, $C$, to account for all possible antiderivatives:
$$ \frac{3x^2}{2} + C $$
The final answer is $\frac{3x^2}{2} + C$
More Information
In indefinite integrals, $C$ represents the constant of integration, accounting for all possible antiderivatives.
Tips
A common mistake is forgetting to add the constant of integration $C$ at the end of the process.