We are tasked with evaluating the integral:

$I = \int_0^1 \frac{16x}{8x^2 + 2} \, dx$

Step 1: Simplifying the expression

Factor out constants from the denominator:

$8x^2 + 2 = 2(4x^2 + 1)$

Thus, the integral becomes:

$I = \int_0^1 \frac{16x}{2(4x^2 + 1)} \, dx = 8 \int_0^1 \frac{x}{4x^2 + 1} \, dx$

Step 2: Substitution

Let’s perform a substitution to simplify the integral further. Set:

$u = 4x^2 + 1$ $\frac{du}{dx} = 8x \quad \Rightarrow \quad dx = \frac{du}{8x}$

Now, substitute into the integral:

$I = 8 \int_{u(0)}^{u(1)} \frac{x}{u} \cdot \frac{du}{8x} = \int_1^5 \frac{du}{u}$

Step 3: Integration

The integral of $\frac{1}{u}$ is $\ln|u|$, so:

$I = \ln|u| \Big|_1^5 = \ln(5) - \ln(1) = \ln(5)$

Thus, the value of the integral is:

$I = \ln(5)$

Final Answer:

$\int_0^1 \frac{16x}{8x^2 + 2} \, dx = \ln(5)$

Would you like more details on the steps, or do you have any questions?

Related Questions:

1. What is the geometric interpretation of this integral?
2. How would the result change if the limits of integration were different?
3. Can this integral be solved using partial fractions?
4. How would you evaluate this integral numerically using a calculator or software?
5. What happens if the coefficient of $x^2$ changes in the denominator?

Tip:

Substitution is a powerful tool for simplifying integrals. Always look for patterns that allow variable substitution when you have quadratic terms in the integrand.