We are tasked with evaluating the definite integral:

$\int_{1}^{e^2} \left(3x + \frac{5}{x}\right) \, dx$

Step 1: Break the integral into two separate terms

$\int_{1}^{e^2} \left(3x + \frac{5}{x}\right) \, dx = \int_{1}^{e^2} 3x \, dx + \int_{1}^{e^2} \frac{5}{x} \, dx$

Step 2: Evaluate each integral

1. First term: $\int 3x \, dx = \frac{3x^2}{2}$ So, the definite integral becomes: $\left[\frac{3x^2}{2}\right]_1^{e^2} = \frac{3(e^2)^2}{2} - \frac{3(1)^2}{2} = \frac{3e^4}{2} - \frac{3}{2}$

2. Second term: $\int \frac{5}{x} \, dx = 5 \ln|x|$ The definite integral becomes: $\left[5 \ln|x|\right]_1^{e^2} = 5 \ln(e^2) - 5 \ln(1)$ Since $\ln(e^2) = 2$ and $\ln(1) = 0$, this simplifies to: $5 \cdot 2 - 5 \cdot 0 = 10$

Step 3: Add the results

Combine the results from both terms: $\frac{3e^4}{2} - \frac{3}{2} + 10$ Simplify: $\frac{3e^4}{2} + \frac{17}{2}$

Final Answer:

$\boxed{\frac{3e^4 + 17}{2}}$

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Related Questions:

1. How do you compute the definite integral step-by-step manually?
2. What is the significance of breaking the integral into simpler terms?
3. How does the natural logarithm property $\ln(e^2) = 2$ help simplify calculations?
4. Can this be solved numerically to verify the result?
5. How do definite integrals apply in real-world scenarios?

Tip:

Always split complex integrals into smaller terms to make calculations more manageable.