The given integral is:

$\int \sqrt{x}(2x - 4) \, dx$

Let's solve it step by step.

Step 1: Expand the expression inside the integral

We rewrite the integral: $\int \sqrt{x}(2x - 4) \, dx = \int \left(2x\sqrt{x} - 4\sqrt{x}\right) \, dx$

Step 2: Simplify terms

Recall that $\sqrt{x} = x^{1/2}$. Using the rule for multiplying exponents, $x^a \cdot x^b = x^{a+b}$: $2x\sqrt{x} = 2x^{3/2}, \quad 4\sqrt{x} = 4x^{1/2}$ So the integral becomes: $\int \left(2x^{3/2} - 4x^{1/2}\right) \, dx$

Step 3: Integrate term by term

We use the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1}, \quad n \neq -1$

1. For $2x^{3/2}$: $\int 2x^{3/2} \, dx = 2 \cdot \frac{x^{5/2}}{5/2} = \frac{4x^{5/2}}{5}$

2. For $-4x^{1/2}$: $\int -4x^{1/2} \, dx = -4 \cdot \frac{x^{3/2}}{3/2} = -\frac{8x^{3/2}}{3}$

Step 4: Combine results

The integral becomes: $\frac{4x^{5/2}}{5} - \frac{8x^{3/2}}{3} + C$

Final Answer:

$\int \sqrt{x}(2x - 4) \, dx = \frac{4x^{5/2}}{5} - \frac{8x^{3/2}}{3} + C$

Let me know if you'd like more details or a graph of this function. Additionally, here are five follow-up questions for practice:

1. How would the integration change if the bounds were specified?
2. What happens if $2x - 4$ is replaced by another polynomial?
3. How can we verify this solution using differentiation?
4. How would this integral look if expressed in terms of a definite integral from $x=0$ to $x=1$?
5. Can this integral be solved using substitution instead?

Tip:

Always simplify expressions before integrating to make the process easier and reduce errors!