We are given two integrals in the problem:

$\int_1^5 f(x) \, dx = 20$ and $\int_4^5 f(x) \, dx = 3.6.$

We are asked to find the value of:

$\int_1^4 f(x) \, dx.$

To solve this, we can use the following property of definite integrals:

$\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.$

For this case:

$\int_1^5 f(x) \, dx = \int_1^4 f(x) \, dx + \int_4^5 f(x) \, dx.$

Substituting the known values:

$20 = \int_1^4 f(x) \, dx + 3.6.$

Now, solve for $\int_1^4 f(x) \, dx$:

$\int_1^4 f(x) \, dx = 20 - 3.6 = 16.4.$

Thus, the value of $\int_1^4 f(x) \, dx$ is 16.4.

Would you like more details, or do you have any further questions?

Here are 5 related questions to expand on this topic:

1. How can the properties of definite integrals help in simplifying complex integral calculations?
2. Can we generalize the approach used here for integrals with different limits?
3. How would the result change if the limits of integration were swapped?
4. What happens when we deal with discontinuous functions in definite integrals?
5. How does the Fundamental Theorem of Calculus apply in solving definite integrals?

Tip: Always break down complex integral problems using the linearity property and properties of definite integrals.