The given integral is:

$\int_0^2 \frac{\log_2(x + 2)}{x + 2} \, dx$

This is a definite integral involving a logarithmic function with base 2. To solve this, we can make use of substitution and known properties of logarithms. Let's proceed step-by-step to find the solution.

Step 1: Substitution

We substitute $u = x + 2$, so that $du = dx$, and the limits of integration change accordingly:

• When $x = 0$, $u = 2$
• When $x = 2$, $u = 4$

Thus, the integral becomes:

$\int_2^4 \frac{\log_2(u)}{u} \, du$

Step 2: Solve Using Change of Logarithm Base Formula

We can rewrite the logarithm in base 2 as a natural logarithm using the change of base formula:

$\log_2(u) = \frac{\ln(u)}{\ln(2)}$

Now the integral becomes:

$\frac{1}{\ln(2)} \int_2^4 \frac{\ln(u)}{u} \, du$

Step 3: Recognize Standard Integral

The integral $\int \frac{\ln(u)}{u} \, du$ is a known standard integral, whose solution is:

$\int \frac{\ln(u)}{u} \, du = \frac{(\ln(u))^2}{2}$

Thus, we have:

$\frac{1}{\ln(2)} \left[ \frac{(\ln(u))^2}{2} \right]_2^4$

Step 4: Evaluate the Integral

Now, substitute the limits of integration:

$\frac{1}{\ln(2)} \left( \frac{(\ln(4))^2}{2} - \frac{(\ln(2))^2}{2} \right)$

Since $\ln(4) = 2\ln(2)$ and $\ln(2)$ is the natural logarithm of 2, simplify:

$\frac{1}{\ln(2)} \left( \frac{(2\ln(2))^2}{2} - \frac{(\ln(2))^2}{2} \right)$

This simplifies to:

$\frac{1}{\ln(2)} \left( \frac{4(\ln(2))^2}{2} - \frac{(\ln(2))^2}{2} \right) = \frac{1}{\ln(2)} \cdot \frac{3(\ln(2))^2}{2}$

Finally, canceling $\ln(2)$:

$\frac{3\ln(2)}{2}$

Thus, the value of the integral is:

$\boxed{\frac{3\ln(2)}{2}}$

Would you like further details or clarifications on any of the steps? Here are 5 related questions to expand the topic:

1. How does substitution simplify integrals in general?
2. Why is the change of base formula important for logarithmic functions?
3. Can we apply the same method to logarithms with different bases?
4. What are other standard integrals involving logarithmic functions?
5. How can we generalize this approach for other types of definite integrals?

Tip: The change of base formula for logarithms is very useful in simplifying integrals where the logarithmic base isn't the natural log. Always look for simplifications!