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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?

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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.

Evaluasi integral berikut: ∫(1 - x)√x dx

Learn how to solve the integral ∫(1 - x)√x dx step by step using the power rule and simplifying radical functions. This solution is designed for advanced high school or early college students studying integral calculus.

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