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(5z^{-3} + 2z^2 + \frac{4}{z^3} – 5)dz}[/latex]

Step 1:

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

[latex]\large{\int(5z^{-3}+2z^2 + \frac{4}{z^3} – 5)dz}[/latex]

[latex]\small{ = \int(5z^{-3})dz +\int(2z^2)dz + \int(\frac{4}{z^3})dz – \int(5)dz}[/latex]

Step 2:

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

[latex]\large{\int(5z^{-3})dz = -\frac{5}{2z^2}}[/latex]

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

[latex]\large{\int(\frac{4}{z^3})dz = -\frac{2}{z^2}}[/latex]

[latex]\large{\int(5)dz = 5z}[/latex]

Step 3:

Now, combining everything, we have:

[latex]\large{\int(5z^{-3}+2z^2+\frac{4}{z^3} – 5)dz }[/latex]

[latex]\large{= \frac{2}{3}z^3 – \frac{5}{2z^2} -\frac{2}{z^2} -5z + C}[/latex]

Finally, simplifying gives us:

[latex]\large{ = \frac{2}{3}z^3 – 5z – \frac{9}{2z^2} + C}[/latex]

Solution Complete!