Question: Given $\int_1^3 f(x) \, dx = 7$, evaluate $\int_1^3 (4f(x) - 3) \, dx$.
1. **State the problem:** We are given the integral $$\int_1^3 f(x) \, dx = 7$$ and need to evaluate $$\int_1^3 (4f(x) - 3) \, dx$$.
2. **Recall the properties of integrals:** The integral of a sum is the sum of the integrals, and constants can be factored out:
$$\int_a^b (c \cdot g(x) + d) \, dx = c \int_a^b g(x) \, dx + d \int_a^b 1 \, dx$$
3. **Apply these properties:**
$$\int_1^3 (4f(x) - 3) \, dx = 4 \int_1^3 f(x) \, dx - 3 \int_1^3 1 \, dx$$
4. **Evaluate the known integrals:**
We know $$\int_1^3 f(x) \, dx = 7$$.
Also, $$\int_1^3 1 \, dx = 3 - 1 = 2$$.
5. **Substitute values:**
$$= 4 \times 7 - 3 \times 2$$
6. **Calculate:**
$$= 28 - 6 = 22$$
**Final answer:** $$\int_1^3 (4f(x) - 3) \, dx = 22$$.