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

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

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

From the problem: $\int_1^5 f(x) \, dx = 14$ $\int_4^5 f(x) \, dx = 5.8$

Substitute these values: $14 = \int_1^4 f(x) \, dx + 5.8$

Solve for $\int_1^4 f(x) \, dx$: $\int_1^4 f(x) \, dx = 14 - 5.8 = 8.2$

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

$\boxed{8.2}$

Let me know if you'd like a detailed explanation or have additional questions!

Related Questions:

1. How does the property of definite integrals work for overlapping intervals?
2. What are the key differences between definite and indefinite integrals?
3. Can you explain how this property applies to non-continuous functions?
4. How can we approximate the value of a definite integral if the function is unknown?
5. Why is it important to have clear bounds when solving definite integrals?

Tip:

Always visualize the bounds of integration on a number line to ensure proper subtraction and application of integral properties.