Certainly! Let's solve the integral
[tex]\[
\int \left( \frac{x^2}{\sqrt{16 + x^2}} + x \right) \, dx
\][/tex]
step-by-step.
### Step 1: Break the integral into two parts.
We can separate the integral into two simpler integrals:
[tex]\[
\int \left( \frac{x^2}{\sqrt{16 + x^2}} \right) dx + \int x \, dx
\][/tex]
### Step 2: Solve the simpler integral [tex]\(\int x \, dx\)[/tex].
This is a basic integral:
[tex]\[
\int x \, dx = \frac{x^2}{2} + C_1
\][/tex]
where [tex]\(C_1\)[/tex] is the constant of integration for this part of the integral.
### Step 3: Solve the integral [tex]\(\int \frac{x^2}{\sqrt{16 + x^2}} \, dx\)[/tex].
For the first integral, we proceed by making an appropriate substitution. Let’s set [tex]\(16 + x^2 = u\)[/tex], hence [tex]\(du = 2x \, dx \)[/tex]. Solving for [tex]\(dx\)[/tex], we get:
[tex]\[
dx = \frac{du}{2x}
\][/tex]
Then, substituting in the integral:
[tex]\[
\int \frac{x^2}{\sqrt{16 + x^2}} \, dx = \int \frac{x^2}{\sqrt{u}} \cdot \frac{du}{2x}
\][/tex]
Simplifying:
[tex]\[
\int \frac{x^2}{\sqrt{u}} \cdot \frac{du}{2x} = \frac{1}{2} \int \frac{x}{\sqrt{u}} \, du
\][/tex]
Since [tex]\(x^2 = u - 16\)[/tex]:
[tex]\[
x = \sqrt{u - 16}
\][/tex]
Thus, the integral becomes:
[tex]\[
\frac{1}{2} \int \frac{\sqrt{u - 16}}{\sqrt{u}} \, du = \frac{1}{2} \int \sqrt{\frac{u - 16}{u}} \, du = \frac{1}{2} \int \sqrt{1 - \frac{16}{u}} \, du
\][/tex]
However, to solve the integral up to its final form, one might involve more complex integration techniques, including trigonometric substitution. For this solution, we use the known result:
[tex]\[
\int \frac{x^2}{\sqrt{16 + x^2}} \, dx = \frac{x \sqrt{16 + x^2}}{2} + 8 \sinh^{-1}\left(\frac{x}{4}\right) + C_2
\][/tex]
### Step 4: Combine the results.
Combining the results of our two integrals, we have:
[tex]\[
\int \left( \frac{x^2}{\sqrt{16 + x^2}} + x \right) \, dx = \left( \frac{x^2}{2} + \frac{x \sqrt{16 + x^2}}{2} - 8 \sinh^{-1}\left(\frac{x}{4}\right) + C_2 \right)
\][/tex]
### Step 5: Combine constants.
The constants [tex]\(C_1\)[/tex] and [tex]\(C_2\)[/tex] combine to form a general constant of integration [tex]\(C\)[/tex]:
[tex]\[
\frac{x^2}{2} + \frac{x \sqrt{16 + x^2}}{2} - 8 \sinh^{-1}\left(\frac{x}{4}\right) + C
\][/tex]
### Final Answer:
[tex]\[
\int \left( \frac{x^2}{\sqrt{16 + x^2}} + x \right) \, dx = \frac{x^2}{2} + \frac{x \sqrt{16 + x^2}}{2} - 8 \sinh^{-1}\left(\frac{x}{4}\right) + C
\][/tex]
This completes the integration process for the given integral.
