Question

the values of several definite integrals are given as follows:
\\(\int_{0}^{2} f(x) dx = 5\\) \\(\int_{0}^{3} f(x) dx = 7\\) \\(\int_{0}^{2} g(x) dx = -3\\) \\(\int_{2}^{3} g(x) dx = 5\\)
use these values and properties of definite integrals to evaluate the indicated definite integral.
\\(\int_{2}^{3} (3f(x) + 2g(x)) dx\\)

Explanation:

Step1: Use integral properties

$$\int_{2}^{3}(3f(x)+2g(x))dx = 3\int_{2}^{3}f(x)dx+2\int_{2}^{3}g(x)dx$$

Step2: Find $\int_{2}^{3}f(x)dx$

$$\int_{0}^{3}f(x)dx=\int_{0}^{2}f(x)dx+\int_{2}^{3}f(x)dx$$
$$7 = 5+\int_{2}^{3}f(x)dx$$
$$\int_{2}^{3}f(x)dx=7 - 5=2$$

Step3: Substitute values

Substitute $\int_{2}^{3}f(x)dx = 2$ and $\int_{2}^{3}g(x)dx = 5$ into $3\int_{2}^{3}f(x)dx+2\int_{2}^{3}g(x)dx$
$$3\times2+2\times5=6 + 10$$

Answer:

$16$