To solve the integral of $g(x) = \frac{x}{2x^2 + 5}$ from 0 to 1, we will follow these steps:

Step 1: Set up the integral

$\int_0^1 \frac{x}{2x^2 + 5} \, dx$

Step 2: Use substitution

Let’s perform a substitution to simplify the integral. Define: $u = 2x^2 + 5$ Now, compute $du$ in terms of $dx$: $\frac{du}{dx} = 4x \quad \Rightarrow \quad du = 4x \, dx$ Thus, $x \, dx = \frac{du}{4}$.

Step 3: Change the limits of integration

When $x = 0$: $u = 2(0)^2 + 5 = 5$ When $x = 1$: $u = 2(1)^2 + 5 = 7$

Step 4: Substitute into the integral

Substituting $u$ and $du$ into the integral: $\int_0^1 \frac{x}{2x^2 + 5} \, dx = \int_5^7 \frac{1}{4} \cdot \frac{1}{u} \, du$

Step 5: Simplify and integrate

Now, we simplify the integral: $\frac{1}{4} \int_5^7 \frac{1}{u} \, du$ The integral of $\frac{1}{u}$ is $\ln|u|$, so we have: $\frac{1}{4} \left[ \ln|u| \right]_5^7 = \frac{1}{4} \left( \ln(7) - \ln(5) \right)$

Step 6: Final answer

Simplifying: $\frac{1}{4} \ln\left(\frac{7}{5}\right)$

Thus, the value of the integral is: $\boxed{\frac{1}{4} \ln\left(\frac{7}{5}\right)}$

Would you like more details on any part of this solution?

Related Questions:

1. How do substitution methods work in integration?
2. What happens if you use integration by parts instead of substitution here?
3. How can you solve integrals involving rational functions?
4. What are the properties of logarithmic functions that make them useful in integration?
5. How would the solution change if the limits of integration were different?

Tip: Always check for substitutions that simplify complex expressions, especially in rational functions. This can make an otherwise difficult integral much more manageable.