The problem asks to evaluate the following integral:

$\int (1 - x) \sqrt{x} \, dx$

Let's solve this step by step:

Step 1: Simplify the integrand

We can distribute $\sqrt{x}$ over the terms inside the parenthesis:

$(1 - x) \sqrt{x} = \sqrt{x} - x \sqrt{x}$

Step 2: Write the integral in a simpler form

Now, we can break the integral into two parts:

$\int (\sqrt{x} - x\sqrt{x}) \, dx = \int \sqrt{x} \, dx - \int x \sqrt{x} \, dx$

Step 3: Rewrite $\sqrt{x}$ as $x^{1/2}$

This will make the integrals easier to handle:

$\int x^{1/2} \, dx - \int x^{3/2} \, dx$

Step 4: Apply the power rule of integration

For both terms, we apply the power rule for integrals, which states:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

So, applying this rule:

1. $\int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}$
2. $\int x^{3/2} \, dx = \frac{x^{5/2}}{5/2} = \frac{2}{5} x^{5/2}$

Step 5: Combine the results

Now, substituting the results into the original expression:

$\frac{2}{3} x^{3/2} - \frac{2}{5} x^{5/2} + C$

Thus, the solution to the integral is:

$\boxed{\frac{2}{3} x^{3/2} - \frac{2}{5} x^{5/2} + C}$

Would you like further details or have any questions?

Here are five related questions for practice:

1. How would you solve $\int (1 + x) \sqrt{x} \, dx$?
2. Can you evaluate $\int \sqrt{x^2 + 1} \, dx$?
3. What is the integral of $\int x^n \, dx$ for general $n$?
4. How do you approach integrals involving trigonometric functions like $\int \sin(x) \sqrt{x} \, dx$?
5. How would you solve a definite version of this integral, such as $\int_0^1 (1 - x) \sqrt{x} \, dx$?

Tip: When solving integrals involving radicals, converting them into exponents can make the process easier to handle.