To solve this integral:

$\int \frac{x}{\sqrt{8x^2 + 5}} \, dx$

we can use a substitution method. Here’s the step-by-step process:

1. Choose a substitution: Let $u = 8x^2 + 5$. Then, $du = 16x \, dx$, or equivalently, $\frac{du}{16} = x \, dx$.

2. Rewrite the integral: Substitute $u = 8x^2 + 5$ and $x \, dx = \frac{du}{16}$: $\int \frac{x}{\sqrt{8x^2 + 5}} \, dx = \int \frac{\frac{du}{16}}{\sqrt{u}} = \frac{1}{16} \int u^{-\frac{1}{2}} \, du$

3. Integrate with respect to $u$: $\frac{1}{16} \int u^{-\frac{1}{2}} \, du = \frac{1}{16} \cdot 2 u^{\frac{1}{2}} = \frac{1}{8} \sqrt{u}$

4. Substitute back $u = 8x^2 + 5$: $\frac{1}{8} \sqrt{8x^2 + 5}$

5. Final Answer: $\int \frac{x}{\sqrt{8x^2 + 5}} \, dx = \frac{1}{8} \sqrt{8x^2 + 5} + C$

Let me know if you'd like further clarification on any steps!

Related Questions

1. What are the steps to integrate functions using substitution?
2. How do you determine which substitution to use for a given integral?
3. Can trigonometric substitutions be used for integrals of this type?
4. What is the difference between definite and indefinite integrals?
5. How does integration relate to finding areas under curves?

Tip

When using substitution in integration, always remember to change both the integrand and the differential $dx$ in terms of the new variable.