3. (e) It is known that $$ \int_{1}^{5} f(x) d x=5 $$, and $$ \int_{3}^{5} 2 f(x) d x=2 $$, what does $$ \int_{1}^{3}(f(x)-2) d x $$ equal to ? A 1 B 3 C 4 D 0

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Given:
$$
\int_{1}^{5} f(x) \, dx = 5
$$
and
$$
\int_{3}^{5} 2f(x) \, dx = 2.
$$

**Step 1: Find $$\int_{3}^{5} f(x) \, dx$$.**

Since the integral is linear, we have:
$$
\int_{3}^{5} 2f(x) \, dx = 2 \int_{3}^{5} f(x) \, dx.
$$
Therefore, we can solve for $$\int_{3}^{5} f(x) \, dx$$:
$$
2 \int_{3}^{5} f(x) \, dx = 2 \quad \Rightarrow \quad \int_{3}^{5} f(x) \, dx = \frac{2}{2} = 1.
$$

**Step 2: Compute $$\int_{1}^{3} f(x) \, dx$$.**

Using the property of integrals over adjacent intervals:
$$
\int_{1}^{5} f(x) \, dx = \int_{1}^{3} f(x) \, dx + \int_{3}^{5} f(x) \, dx,
$$
thus:
$$
5 = \int_{1}^{3} f(x) \, dx + 1 \quad \Rightarrow \quad \int_{1}^{3} f(x) \, dx = 5 - 1 = 4.
$$

**Step 3: Evaluate $$\int_{1}^{3} (f(x) - 2) \, dx$$.**

The integral can be separated as:
$$
\int_{1}^{3} (f(x) - 2) \, dx = \int_{1}^{3} f(x) \, dx - \int_{1}^{3} 2 \, dx.
$$
We already found:
$$
\int_{1}^{3} f(x) \, dx = 4.
$$
And the integral of the constant function $$2$$ is:
$$
\int_{1}^{3} 2 \, dx = 2(3 - 1) = 4.
$$
Thus:
$$
\int_{1}^{3} (f(x) - 2) \, dx = 4 - 4 = 0.
$$

The correct answer is D $$0$$.
The integral $$ \int_{1}^{3} (f(x) - 2) \, dx $$ equals 0.

3. (e) It is known that \( \int_{1}^{5} f(x) d x=5 \), and \( \int_{3}^{5} 2 f(x) d x=2 \), what does \( \int_{1}^{3}(f(x)-2) d x \) equal to ? A 1 B 3 C 4 D 0

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