The image shows three integral equations involving the function $f(x)$:

1. $\int_1^3 (f(x) + 2) \, dx = 2$
2. $\int_3^4 (f(x) - 3) \, dx = 5$
3. $\int_1^4 f(x) \, dx = 3$

We can break these down step by step to solve for the integral of $f(x)$ over different intervals.

Step 1: Solve the first equation

The first equation is:

$\int_1^3 (f(x) + 2) \, dx = 2$

We can split the integral:

$\int_1^3 f(x) \, dx + \int_1^3 2 \, dx = 2$

The second integral is straightforward:

$\int_1^3 2 \, dx = 2 \times (3 - 1) = 4$

Now, substituting into the equation:

$\int_1^3 f(x) \, dx + 4 = 2$

Solving for $\int_1^3 f(x) \, dx$:

$\int_1^3 f(x) \, dx = 2 - 4 = -2$

Step 2: Solve the second equation

The second equation is:

$\int_3^4 (f(x) - 3) \, dx = 5$

Split the integral:

$\int_3^4 f(x) \, dx - \int_3^4 3 \, dx = 5$

The second integral is:

$\int_3^4 3 \, dx = 3 \times (4 - 3) = 3$

Substituting into the equation:

$\int_3^4 f(x) \, dx - 3 = 5$

Solving for $\int_3^4 f(x) \, dx$:

$\int_3^4 f(x) \, dx = 5 + 3 = 8$

Step 3: Use the third equation

The third equation is:

$\int_1^4 f(x) \, dx = 3$

We can split the integral:

$\int_1^4 f(x) \, dx = \int_1^3 f(x) \, dx + \int_3^4 f(x) \, dx$

Substitute the values we found earlier:

$3 = (-2) + 8$

This holds true, confirming our results are consistent.

Summary of Results:

• $\int_1^3 f(x) \, dx = -2$
• $\int_3^4 f(x) \, dx = 8$
• $\int_1^4 f(x) \, dx = 3$

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

Here are some related questions:

1. How do we evaluate integrals involving constants?
2. What is the fundamental theorem of calculus, and how does it apply here?
3. How can we use substitution to simplify more complex integrals?
4. Can you explain the relationship between definite integrals and areas?
5. What happens if $f(x)$ is piecewise in these problems?

Tip: When splitting integrals, remember that you can always break them up over adjacent intervals and solve each part separately.