To evaluate the integral

$\int_{3}^{9} \left(2f(x) + 1\right) \, dx$

we use the properties of integrals:

1. Linearity of Integrals:
   $\int_{a}^{b} \left(cf(x) + g(x)\right) \, dx = c\int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx$

Given: $\int_{3}^{9} f(x) \, dx = 7$

Step 1: Break the integral into two parts:

$\int_{3}^{9} \left(2f(x) + 1\right) \, dx = \int_{3}^{9} 2f(x) \, dx + \int_{3}^{9} 1 \, dx$

Step 2: Evaluate each term:

• For $\int_{3}^{9} 2f(x) \, dx$: $\int_{3}^{9} 2f(x) \, dx = 2 \int_{3}^{9} f(x) \, dx = 2(7) = 14$

• For $\int_{3}^{9} 1 \, dx$: $\int_{3}^{9} 1 \, dx = \text{length of the interval} = 9 - 3 = 6$

Step 3: Combine the results:

$\int_{3}^{9} \left(2f(x) + 1\right) \, dx = 14 + 6 = 20$

Final Answer:

$\int_{3}^{9} \left(2f(x) + 1\right) \, dx = 20$

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Here are 5 related questions to explore further:

1. How does linearity of integrals simplify calculations?
2. Can we generalize this approach to any constant multiplied with $f(x)$?
3. What happens if the interval bounds are reversed (e.g., $9$ to $3$)?
4. How do definite and indefinite integrals differ in their interpretation?
5. How can integrals be approximated if the function $f(x)$ is unknown?

Tip: Always check the properties of integrals when dealing with combinations of functions. It simplifies calculations significantly!