If `int_(-3)^(2)f(x) dx= 7/3` and `int_(-3)^(9) f(x) dx = -5/6` , then what is the value of `int_(2)^(9)f(x) dx` ?

Options

A. `-(19)/(6)`
B. `(19)/(6)`
C. `3/2`
D. `-(3)/(2)`

Explanation

To find the value of \( \int_{2}^{9} f(x) \, dx \), we can use the properties of definite integrals. We have the following information:

1. \( \int_{-3}^{2} f(x) \, dx = \frac{7}{3} \)
2. \( \int_{-3}^{9} f(x) \, dx = -\frac{5}{6} \)

We can express \( \int_{2}^{9} f(x) \, dx \) in terms of the given integrals using the property:

\[ \int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx \]

Here, we can set \( a = -3 \), \( b = 2 \), and \( c = 9 \). Thus, we can rewrite:

\[ \int_{-3}^{9} f(x) \, dx = \int_{-3}^{2} f(x) \, dx + \int_{2}^{9} f(x) \, dx \]

From this, we can isolate \( \int_{2}^{9} f(x) \, dx \):

\[ \int_{2}^{9} f(x) \, dx = \int_{-3}^{9} f(x) \, dx - \int_{-3}^{2} f(x) \, dx \]

Now, substituting the known values:

\[ \int_{2}^{9} f(x) \, dx = -\frac{5}{6} - \frac{7}{3} \]

To perform the subtraction, we need a common denominator. The least common multiple of 6 and 3 is 6. We can rewrite \( \frac{7}{3} \) as \( \frac{14}{6} \):

\[ \int_{2}^{9} f(x) \, dx = -\frac{5}{6} - \frac{14}{6} \]

Now, combine the fractions:

\[ \int_{2}^{9} f(x) \, dx = -\frac{5 + 14}{6} = -\frac{19}{6} \]

Thus, the value of \( \int_{2}^{9} f(x) \, dx \) is:

\[ \boxed{-\frac{19}{6}} \]