To solve the problem shown in the image:

We are given:

$\int_{1}^{5} f(x) \, dx = 8$ $\int_{4}^{5} f(x) \, dx = 4$

We need to find:

$\int_{1}^{4} f(x) \, dx$

Solution:

The property of definite integrals states that if $a < b < c$, then: $\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx$

Using this property, we can write: $\int_{1}^{5} f(x) \, dx = \int_{1}^{4} f(x) \, dx + \int_{4}^{5} f(x) \, dx$

Substitute the known values: $8 = \int_{1}^{4} f(x) \, dx + 4$

Solve for $\int_{1}^{4} f(x) \, dx$: $\int_{1}^{4} f(x) \, dx = 8 - 4 = 4$

Final Answer:

$\boxed{4}$

Would you like more explanation, or do you have further questions?

Related Questions:

1. How does the property of splitting integrals work for overlapping intervals?
2. What happens if one of the integrals is negative?
3. Can definite integrals be used for area under the curve in this problem?
4. How can we generalize this solution to other limits?
5. What happens when the integral limits are reversed?

Tip:

Always check for integral properties when given multiple conditions—they simplify the problem-solving process.