We are given the derivative function and asked to find the original function. Thus, we must integrate the derivative function, as follows:

[latex]\large{\int(3x^4-2x^3-15x^2+3x-1)dx}[/latex]

Step 1:

We can first apply the integration to each term as follows:

[latex]\large{\int(3x^4-2x^3-15x^2+3x-1)dx}[/latex]

[latex]\scriptsize{ = \int(3x^4)dx -\int(2x^3)dx – \int(15x^2)dx + \int(3x)dx – \int(1)dx}[/latex]

Step 2:

Now, we can use the integral power rule to evaluate each integral individually, as shown:

[latex]\large{\int(3x^4)dx = \frac{3}{5}x^5}[/latex]

[latex]\large{\int(2x^3)dx = \frac{1}{2}x^4}[/latex]

[latex]\large{\int(15x^2)dx = 5x^3 }[/latex]

[latex]\large{\int(3x)dx = \frac{3}{2}x^2}[/latex]

[latex]\large{\int(1)dx = x }[/latex]

Step 3:

Now, combining everything, we have:

[latex]\large{\int(3x^4-2x^3-15x^2+3x-1)dx}[/latex]

[latex]\large{= \frac{3}{5}x^5 – \frac{1}{2}x^4 – 5x^3 + \frac{3}{2}x^2 – x + C }[/latex]

Solution Complete!