If it is known that `int _(0) ^(10) f (x ) dx =17 and int _(0) ^(8) f (x ) dx =12, ` find` int _(8)^(10) f (x) dx` , given f(x) is continuous.

Answer

To solve the problem, we need to find the value of the integral from 8 to 10 of the function \( f(x) \) given the following information:

1. \( \int_{0}^{10} f(x) \, dx = 17 \)
2. \( \int_{0}^{8} f(x) \, dx = 12 \)

We can use the property of definite integrals that allows us to break the integral over a larger interval into the sum of integrals over smaller intervals. Specifically, we can express the integral from 0 to 10 as the sum of the integrals from 0 to 8 and from 8 to 10:

\[ \int_{0}^{10} f(x) \, dx = \int_{0}^{8} f(x) \, dx + \int_{8}^{10} f(x) \, dx \]

Now, we can substitute the known values into this equation:

\[ 17 = 12 + \int_{8}^{10} f(x) \, dx \]

Next, we can isolate the integral \( \int_{8}^{10} f(x) \, dx \):

\[ \int_{8}^{10} f(x) \, dx = 17 - 12 \]

Calculating the right side gives us:

\[ \int_{8}^{10} f(x) \, dx = 5 \]

Thus, the value of the integral from 8 to 10 is:

\[ \int_{8}^{10} f(x) \, dx = 5 \]

### Final Answer:
\[ \int_{8}^{10} f(x) \, dx = 5 \]