2pi*integral(0to3)sqrt(2x)(sqrt((x+1)/x))

We are given the following expression to solve:

\[
2\pi \int_0^3 \sqrt{2x} \cdot \sqrt{\frac{x+1}{x}} \, dx
\]

Let's break this down step by step.

### Step 1: Simplifying the integrand
We can simplify the integrand \( \sqrt{2x} \cdot \sqrt{\frac{x+1}{x}} \) by combining the square roots:

\[
\sqrt{2x} \cdot \sqrt{\frac{x+1}{x}} = \sqrt{2x \cdot \frac{x+1}{x}} = \sqrt{2(x+1)}
\]

Thus, the integral becomes:

\[
2\pi \int_0^3 \sqrt{2(x+1)} \, dx
\]

### Step 2: Substitution
Let’s perform a substitution to simplify the integral further. Let:

\[
u = x + 1 \quad \text{which implies} \quad du = dx
\]

Also, when \( x = 0 \), \( u = 1 \), and when \( x = 3 \), \( u = 4 \).

Substituting into the integral:

\[
2\pi \int_1^4 \sqrt{2u} \, du
\]

### Step 3: Evaluating the integral
We can now factor out \( \sqrt{2} \) and integrate:

\[
2\pi \cdot \sqrt{2} \int_1^4 \sqrt{u} \, du = 2\pi \cdot \sqrt{2} \int_1^4 u^{1/2} \, du
\]

Now, integrate \( u^{1/2} \):

\[
\int u^{1/2} \, du = \frac{2}{3} u^{3/2}
\]

So the integral becomes:

\[
2\pi \cdot \sqrt{2} \cdot \frac{2}{3} \left[ u^{3/2} \right]_1^4
\]

### Step 4: Plugging in the limits
Now, substitute the limits \( u = 4 \) and \( u = 1 \):

\[
\frac{2}{3} \left( 4^{3/2} - 1^{3/2} \right) = \frac{2}{3} \left( 8 - 1 \right) = \frac{2}{3} \times 7 = \frac{14}{3}
\]

### Step 5: Final answer
Finally, multiply by \( 2\pi \cdot \sqrt{2} \):

\[
2\pi \cdot \sqrt{2} \cdot \frac{14}{3} = \frac{28\pi\sqrt{2}}{3}
\]

Thus, the solution to the integral is:

\[
\boxed{\frac{28\pi\sqrt{2}}{3}}
\]

Would you like more details or have any questions?

### Related questions:
1. How would the result change if the limits of integration were different?
2. Can this integral be solved using another method?
3. What happens if you change the function inside the square roots?
4. How does substitution help simplify integrals in general?
5. What is the geometric interpretation of this integral?

### Tip:
For integrals involving square roots, try substitution to simplify the expression.

Learn how to solve the integral 2pi * integral(0 to 3) sqrt(2x) * sqrt((x+1)/x) using substitution and step-by-step simplification. The solution involves applying square roots, substitution methods, and the fundamental theorem of calculus. Ideal for college-level calculus students.

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