The problem is asking for the indefinite integral of the expression:

$\int \sqrt{1 + 2x^2} \cdot (4x) \, dx$

To solve this, let's go through it step-by-step.

Step 1: Identify a Substitution

The expression inside the square root, $1 + 2x^2$, suggests using a substitution that simplifies it. Let: $u = 1 + 2x^2$

Then, differentiating with respect to $x$: $\frac{du}{dx} = 4x \quad \Rightarrow \quad dx = \frac{du}{4x}$

Step 2: Substitute in the Integral

Rewrite the integral in terms of $u$: $\int \sqrt{u} \cdot (4x) \cdot \frac{dx}{4x}$

The $4x$ terms cancel out, so we have: $\int \sqrt{u} \, du$

Step 3: Integrate with Respect to $u$

The integral now simplifies to: $\int u^{1/2} \, du$

Using the power rule for integration, we get: $\int u^{1/2} \, du = \frac{u^{3/2}}{\frac{3}{2}} = \frac{2}{3} u^{3/2} + C$

Step 4: Substitute Back for $u$

Recall that $u = 1 + 2x^2$, so substitute back: $\frac{2}{3} (1 + 2x^2)^{3/2} + C$

Final Answer

The indefinite integral is: $\int \sqrt{1 + 2x^2} \cdot (4x) \, dx = \frac{2}{3} (1 + 2x^2)^{3/2} + C$

Would you like more details on any step, or have any questions?

Here are some related questions you may find interesting:

1. How does the power rule for integration work?
2. How do we choose a substitution in integrals?
3. What are some other common integral techniques?
4. How does the square root affect integration?
5. Why do we add a constant $C$ after integrating?

Tip: For integrals involving square roots, look for substitutions that can simplify the expression inside the root!