The given integral is:

$\int x \sqrt{5 + 2x^2} \, dx$

Solution:

To solve this, we use substitution. Let:

$u = 5 + 2x^2$

Then:

$\frac{du}{dx} = 4x \quad \implies \quad du = 4x \, dx$

From this, we have:

$x \, dx = \frac{1}{4} du$

Substituting into the integral:

$\int x \sqrt{5 + 2x^2} \, dx = \int \sqrt{u} \cdot \frac{1}{4} \, du$

Simplify:

$\int x \sqrt{5 + 2x^2} \, dx = \frac{1}{4} \int u^{1/2} \, du$

Integral of $u^{1/2}$:

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

So:

$\frac{1}{4} \int u^{1/2} \, du = \frac{1}{4} \cdot \frac{2}{3} u^{3/2} = \frac{1}{6} u^{3/2}$

Substitute $u = 5 + 2x^2$ back:

$\int x \sqrt{5 + 2x^2} \, dx = \frac{1}{6} (5 + 2x^2)^{3/2} + C$

Final Answer:

$\int x \sqrt{5 + 2x^2} \, dx = \frac{1}{6} (5 + 2x^2)^{3/2} + C$

Do you have any questions or need further clarification?

Related Questions:

1. What is the role of substitution in solving integrals like this?
2. How do you handle more complex substitutions in definite integrals?
3. Can you demonstrate integration by parts for a similar type of function?
4. What happens if the function inside the square root is negative (i.e., complex analysis)?
5. Can this integral be solved using numerical methods instead of substitution?

Tip:

Always verify your substitution by differentiating the final result to ensure it matches the original integrand!