We are tasked with evaluating the integral:

$I = \int_1^{e^2} \frac{(\ln x)^2}{x} \, dx$

Step 1: Use Substitution

Let’s apply substitution to simplify this integral. Set:

$u = \ln x$

Then, the differential $du$ is given by:

$du = \frac{1}{x} \, dx$

Thus, $\frac{1}{x} \, dx = du$. Now we can rewrite the limits of integration. When $x = 1$, we have:

$u = \ln 1 = 0$

When $x = e^2$, we have:

$u = \ln e^2 = 2$

Thus, the integral becomes:

$I = \int_0^2 u^2 \, du$

Step 2: Evaluate the New Integral

Now we integrate $u^2$:

$\int u^2 \, du = \frac{u^3}{3}$

We can now compute the definite integral:

$I = \left[ \frac{u^3}{3} \right]_0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}$

Final Answer:

The value of the integral is:

$I = \frac{8}{3}$

Would you like a more detailed explanation of any step? Or do you have any other questions?

Further questions:

1. What happens if we use a different substitution, like $u = \ln x / x$?
2. Can this method be applied to more complicated integrals involving logarithms?
3. How do we approach integrals where the logarithmic function is in the numerator or denominator in different forms?
4. What are other common substitutions used in logarithmic integrals?
5. How would you solve integrals involving other powers of logarithms, such as $(\ln x)^3$?

Tip: When dealing with integrals involving logarithms and powers of $x$, try recognizing patterns where substitution simplifies the integral, especially when expressions of the form $\frac{1}{x}$ or logarithmic powers appear.