The integral provided is:

$\int 3x \sqrt{x^2 + 8} \, dx$

Step 1: Choose $u$ and $dv$

Let's use substitution here first to simplify the problem. Let:

$u = x^2 + 8$

This choice makes the square root term easier to handle.

Step 2: Compute $du$

Differentiating $u$ with respect to $x$:

$du = 2x \, dx$

So:

$\frac{1}{2} du = x \, dx$

Step 3: Rewrite the Integral

Substitute back into the integral:

$\int 3x \sqrt{x^2 + 8} \, dx = 3 \int \sqrt{u} \cdot \frac{1}{2} du = \frac{3}{2} \int u^{1/2} \, du$

Step 4: Integrate

Now integrate:

$\frac{3}{2} \int u^{1/2} \, du = \frac{3}{2} \cdot \frac{2}{3} u^{3/2} + C = u^{3/2} + C$

Step 5: Substitute Back

Finally, substitute back $u = x^2 + 8$:

$(x^2 + 8)^{3/2} + C$

Summary

For the substitution method:

• $u$: $x^2 + 8$
• $du$: $2x \, dx$
• $dv$: Not directly used, but you simplified using $du$ and the expression of $\sqrt{u}$.

Would you like further details or have any other questions?

Here are five related questions to explore further:

1. How does substitution simplify integrals involving square roots?
2. What other techniques could you use if substitution doesn't seem straightforward?
3. How can you verify the result of an integral with a square root?
4. What happens if the integral had $x^3 \sqrt{x^2 + 8}$ instead of $3x$?
5. Can you apply integration by parts to this problem, and if so, how would it change the solution?

Tip: When dealing with square roots of quadratic expressions, substitution is usually an effective approach, especially when the expression inside the square root can be simplified.