To solve the definite integral [tex]\(\int \frac{x z + x y}{\sqrt{16 + x^2}} \, dx\)[/tex], let's go through the steps in detail.
1. Combine the Terms in the Numerator:
The integrand is [tex]\(\frac{x z + x y}{\sqrt{16 + x^2}}\)[/tex].
We can factor out the [tex]\(x\)[/tex] in the numerator:
[tex]\[
\frac{x z + x y}{\sqrt{16 + x^2}} = \frac{x(z + y)}{\sqrt{16 + x^2}}
\][/tex]
2. Simplify the Integrand:
Now, the integrand becomes:
[tex]\[
(z + y) \frac{x}{\sqrt{16 + x^2}}
\][/tex]
3. Separate the Constants:
Since [tex]\(z + y\)[/tex] is a constant with respect to [tex]\(x\)[/tex], we can factor it out of the integral:
[tex]\[
(z + y) \int \frac{x}{\sqrt{16 + x^2}} \, dx
\][/tex]
4. Solve the Integral:
We now need to integrate [tex]\(\frac{x}{\sqrt{16 + x^2}}\)[/tex]. This is a standard integral that can be recognized by using a trigonometric substitution or recognizing the derivative pattern. The integral:
[tex]\[
\int \frac{x}{\sqrt{16 + x^2}} \, dx
\][/tex]
is known to be:
[tex]\[
\sqrt{16 + x^2}
\][/tex]
5. Combine Results:
Therefore, the integral we are solving is:
[tex]\[
(z + y) \int \frac{x}{\sqrt{16 + x^2}} \, dx = (z + y) \sqrt{16 + x^2}
\][/tex]
6. Final Answer:
Rewriting the result, we get:
[tex]\[
\int \frac{x z + x y}{\sqrt{16 + x^2}} \, dx = y \sqrt{16 + x^2} + z \sqrt{16 + x^2}
\][/tex]
Hence, the detailed solution to the integral [tex]\(\int \frac{x z + x y}{\sqrt{16 + x^2}} \, dx\)[/tex] is:
[tex]\[
y \sqrt{16 + x^2} + z \sqrt{16 + x^2}
\][/tex]
