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.

$\frac{3x^2}{2} + C$

The final answer is $\frac{3x^2}{2} + C$

Steps to Solve

1. Identify the integral to be solved

We are given the function $3x$ and need to find its indefinite integral with respect to $x$.

$$\int 3x \ dx$$

2. 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, $x^n = x^1$.

3. Multiply by the constant

Since we have a constant 3 multiplying $x$, we can factor out the constant and then apply the integration rule:

$$\int 3x \ dx = 3 \int x \ dx$$

4. Integrate $x$ using the power rule

Use the power rule to integrate $x$:

$$\int x \ dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2}$$

5. Multiply the result by the constant

Now multiply by the constant 3:

$$3 \cdot \frac{x^2}{2} = \frac{3x^2}{2}$$

6. 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 $\frac{3x^2}{2} + C$

In indefinite integrals, $C$ represents the constant of integration, accounting for all possible antiderivatives.
A common mistake is forgetting to add the constant of integration $C$ at the end of the process.