The given integral is:

$\int \frac{x \, dx}{4x^2 + 1}.$

Solution:

Let us solve this step by step:

1. Substitute: Let $u = 4x^2 + 1$. Then: $du = 8x \, dx \quad \text{or} \quad \frac{du}{8} = x \, dx.$

2. Rewriting the integral: Substituting $u$ and $du$ into the integral, we get: $\int \frac{x \, dx}{4x^2 + 1} = \int \frac{1}{u} \cdot \frac{du}{8}.$

3. Simplify: Factor out the constant $\frac{1}{8}$: $\int \frac{x \, dx}{4x^2 + 1} = \frac{1}{8} \int \frac{1}{u} \, du.$

4. Integrate: The integral of $\frac{1}{u}$ is $\ln|u|$: $\frac{1}{8} \int \frac{1}{u} \, du = \frac{1}{8} \ln|u| + C,$ where $C$ is the constant of integration.

5. Back-substitute $u$: Recall that $u = 4x^2 + 1$. Substituting this back, we get: $\frac{1}{8} \ln|u| + C = \frac{1}{8} \ln(4x^2 + 1) + C.$

Final Answer:

$\int \frac{x \, dx}{4x^2 + 1} = \frac{1}{8} \ln(4x^2 + 1) + C.$

Would you like a more detailed explanation or related examples?

Related Questions:

1. What is the integral of $\int \frac{x \, dx}{x^2 + 1}$?
2. How do substitution techniques work for more complex integrals?
3. What is the integral of $\int \frac{dx}{4x^2 + 1}$?
4. Can you explain the difference between substitution and partial fractions in integration?
5. How can you verify the result of an integral using differentiation?

Tip:

Always check if substitution simplifies the integral—it's a powerful first step in solving integrals!