To evaluate $e^2 \int_1^e \frac{3x + 5}{x} e^x \, dx$, let's break it down step by step:

Step 1: Simplify the integrand

The integrand is: $\frac{3x + 5}{x} e^x = (3 + \frac{5}{x}) e^x$ So, the integral becomes: $e^2 \int_1^e \left(3e^x + \frac{5}{x}e^x\right) \, dx$

Step 2: Split the integral

We can separate it into two parts: $e^2 \int_1^e \left(3e^x\right) \, dx + e^2 \int_1^e \left(\frac{5}{x}e^x\right) \, dx$

Step 3: Evaluate the first term

The first term is: $e^2 \int_1^e 3e^x \, dx = 3e^2 \int_1^e e^x \, dx$ The integral of $e^x$ is $e^x$, so: $3e^2 \left[ e^x \right]_1^e = 3e^2 \left(e - e^1\right) = 3e^2(e - 1)$

Step 4: Evaluate the second term

For the second term: $e^2 \int_1^e \frac{5}{x}e^x \, dx$ Here, let $u = e^x$, so $du = e^x dx$ and $x = \ln u$. When $x = 1$, $u = e$; and when $x = e$, $u = e^e$.

Rewriting the integral: $e^2 \int_{e}^{e^e} \frac{5}{\ln u} \, du$ This integral does not simplify to an elementary function, so it is left in this form: $e^2 \int_{e}^{e^e} \frac{5}{\ln u} \, du$

Step 5: Combine results

Thus, the final result is: $3e^2(e - 1) + e^2 \int_{e}^{e^e} \frac{5}{\ln u} \, du$

Let me know if you need further steps, a numerical approximation, or clarification!

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Questions to Explore:

1. What is the significance of rewriting the integral using substitution?
2. Can this integral be solved using numerical techniques?
3. What properties of exponential functions make this integral interesting?
4. How does the integral behave for different limits of integration?
5. How can logarithmic substitution simplify similar integrals?

Tip:

When dealing with exponential and logarithmic terms, substitutions like $u = e^x$ or $u = \ln x$ are often very useful!