Let's solve each part step-by-step using the given integrals:

Given:

1. $\int_{1}^{2} f(x) \, dx = -4$
2. $\int_{1}^{5} f(x) \, dx = 6$
3. $\int_{1}^{5} g(x) \, dx = 8$

We will use these values to find each integral in the question.

Solution for each part:

Part (a)

$\int_{3}^{5} 2g(x) \, dx$ Using the property of integrals, we can factor out constants: $\int_{3}^{5} 2g(x) \, dx = 2 \int_{3}^{5} g(x) \, dx$ To evaluate $\int_{3}^{5} g(x) \, dx$, we use: $\int_{1}^{5} g(x) \, dx = \int_{1}^{3} g(x) \, dx + \int_{3}^{5} g(x) \, dx$ So, $8 = \int_{1}^{3} g(x) \, dx + \int_{3}^{5} g(x) \, dx$ Let $\int_{3}^{5} g(x) \, dx = I$. Then, $8 = \int_{1}^{3} g(x) \, dx + I$ Since we do not have $\int_{1}^{3} g(x) \, dx$, we cannot solve this part directly without further information.

Part (b)

$\int_{5}^{3} g(x) \, dx$ Using the property of reversing limits in integrals: $\int_{5}^{3} g(x) \, dx = -\int_{3}^{5} g(x) \, dx$ Without the value of $\int_{3}^{5} g(x) \, dx$, we cannot solve this directly either.

Part (c)

$\int_{1}^{2} 3f(x) \, dx$ Using the constant multiple rule: $\int_{1}^{2} 3f(x) \, dx = 3 \int_{1}^{2} f(x) \, dx = 3 \cdot (-4) = -12$

Part (d)

$\int_{2}^{5} f(x) \, dx$ We know: $\int_{1}^{5} f(x) \, dx = \int_{1}^{2} f(x) \, dx + \int_{2}^{5} f(x) \, dx$ So, $6 = -4 + \int_{2}^{5} f(x) \, dx$ $\int_{2}^{5} f(x) \, dx = 6 + 4 = 10$

Part (e)

$\int_{1}^{5} \left[g(x) - f(x)\right] \, dx$ Using the linearity of integrals: $\int_{1}^{5} \left[g(x) - f(x)\right] \, dx = \int_{1}^{5} g(x) \, dx - \int_{1}^{5} f(x) \, dx$ $= 8 - 6 = 2$

Part (f)

$\int_{1}^{5} \left[4f(x) - g(x)\right] \, dx$ Again, using linearity: $\int_{1}^{5} \left[4f(x) - g(x)\right] \, dx = 4 \int_{1}^{5} f(x) \, dx - \int_{1}^{5} g(x) \, dx$ $= 4 \cdot 6 - 8 = 24 - 8 = 16$

Summary of Answers

• (a) Requires additional information for $\int_{3}^{5} g(x) \, dx$.
• (b) Depends on the result of (a).
• (c) $-12$
• (d) $10$
• (e) $2$
• (f) $16$

Would you like details on any specific part, or do you have any questions?

Here are five related questions that expand on this information:

1. How would the answers change if $\int_{1}^{3} g(x) \, dx$ were given?
2. Can we derive $\int_{1}^{3} g(x) \, dx$ using any additional assumptions?
3. How do properties of linearity help simplify complex integral expressions?
4. What methods could be used if we need numerical values for unknown integrals?
5. How can we verify these results if given functions $f(x)$ and $g(x)$ explicitly?

Tip: When given partial information on integrals over different intervals, always check if adding or subtracting known values can help fill in the gaps.