integral of 3x

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:

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