Find the indefinite integral of the function
f(z) =  (z^4 + z)(3z − 1) / z^2√z

To find the indefinite integral of the function \( f(z) = \frac{(z^4 + z)(3z − 1)}{z^2\sqrt{z}} \), we will first simplify the integrand and then integrate the resulting expression.

### Step 1: Simplify the function
Let's expand the numerator:

\[
(z^4 + z)(3z - 1) = z^4(3z - 1) + z(3z - 1)
\]

Expanding each term:

\[
z^4(3z - 1) = 3z^5 - z^4
\]
\[
z(3z - 1) = 3z^2 - z
\]

So the numerator becomes:

\[
3z^5 - z^4 + 3z^2 - z
\]

Now, let's divide each term in the numerator by the denominator \( z^2\sqrt{z} = z^{2 + \frac{1}{2}} = z^{\frac{5}{2}} \):

\[
\frac{3z^5}{z^{\frac{5}{2}}} - \frac{z^4}{z^{\frac{5}{2}}} + \frac{3z^2}{z^{\frac{5}{2}}} - \frac{z}{z^{\frac{5}{2}}}
\]

Simplifying each term:

\[
= 3z^{\frac{10}{2} - \frac{5}{2}} - z^{\frac{8}{2} - \frac{5}{2}} + 3z^{\frac{4}{2} - \frac{5}{2}} - z^{\frac{2}{2} - \frac{5}{2}}
\]

This results in:

\[
= 3z^{\frac{5}{2}} - z^{\frac{3}{2}} + 3z^{-\frac{1}{2}} - z^{-\frac{3}{2}}
\]

### Step 2: Integrate each term
Now we integrate each term separately:

\[
\int 3z^{\frac{5}{2}} \, dz - \int z^{\frac{3}{2}} \, dz + \int 3z^{-\frac{1}{2}} \, dz - \int z^{-\frac{3}{2}} \, dz
\]

The integrals of these terms are:

\[
\int z^n \, dz = \frac{z^{n+1}}{n+1} + C
\]

Applying this:

\[
\int 3z^{\frac{5}{2}} \, dz = \frac{3z^{\frac{7}{2}}}{\frac{7}{2}} = \frac{6z^{\frac{7}{2}}}{7}
\]

\[
\int z^{\frac{3}{2}} \, dz = \frac{z^{\frac{5}{2}}}{\frac{5}{2}} = \frac{2z^{\frac{5}{2}}}{5}
\]

\[
\int 3z^{-\frac{1}{2}} \, dz = 3 \cdot \frac{z^{\frac{1}{2}}}{\frac{1}{2}} = 6z^{\frac{1}{2}}
\]

\[
\int z^{-\frac{3}{2}} \, dz = \frac{z^{-\frac{1}{2}}}{-\frac{1}{2}} = -2z^{-\frac{1}{2}}
\]

### Step 3: Combine the results
Combining all the terms together, we get:

\[
\frac{6z^{\frac{7}{2}}}{7} - \frac{2z^{\frac{5}{2}}}{5} + 6z^{\frac{1}{2}} + 2z^{-\frac{1}{2}} + C
\]

### Final Answer:
\[
\int f(z) \, dz = \frac{6z^{\frac{7}{2}}}{7} - \frac{2z^{\frac{5}{2}}}{5} + 6z^{\frac{1}{2}} + 2z^{-\frac{1}{2}} + C
\]

This is the indefinite integral of the given function.

Do you want any details or have any questions? Here are 8 related questions you might find interesting:

1. How do you integrate functions with fractional exponents?
2. What are some common techniques for simplifying complex integrands?
3. How can substitution be used to solve integrals?
4. What is the significance of the constant of integration \( C \)?
5. How do you apply integration by parts?
6. What are improper integrals, and how are they evaluated?
7. How can integration be used to find the area under a curve?
8. What is the difference between definite and indefinite integrals?

**Tip:** Always simplify the integrand as much as possible before integrating, as it can make the integration process easier.

Learn how to find the indefinite integral of the function f(z) = (z^4 + z)(3z − 1) / z^2√z with a detailed step-by-step solution. This problem covers integration of functions with fractional exponents and substitution techniques. Suitable for college-level students.

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