https://file.aiquickdraw.com/mathbot/file/1728942929825xe1m5ywz.png

The integral shown is:

\[
\int g'(x) \sqrt{6 + 10g(x)} \, dx
\]

This can be solved by substitution. Let's use the following substitution:

### Step 1: Substitution
Let:
\[
u = 6 + 10g(x)
\]
Thus, differentiating both sides with respect to \(x\):
\[
du = 10g'(x) \, dx
\]
or equivalently:
\[
\frac{du}{10} = g'(x) \, dx
\]

### Step 2: Rewrite the Integral
Substituting into the integral:
\[
\int g'(x) \sqrt{6 + 10g(x)} \, dx = \int \sqrt{u} \cdot \frac{du}{10}
\]

### Step 3: Solve the Integral
Now, the integral simplifies to:
\[
\frac{1}{10} \int u^{1/2} \, du = \frac{1}{10} \cdot \frac{2}{3} u^{3/2} + C = \frac{1}{15} u^{3/2} + C
\]

### Step 4: Substitute \(u\) back
Recall that \(u = 6 + 10g(x)\), so the solution is:
\[
\frac{1}{15} \left( 6 + 10g(x) \right)^{3/2} + C
\]

Thus, the final result is:
\[
\int g'(x) \sqrt{6 + 10g(x)} \, dx = \frac{1}{15} \left( 6 + 10g(x) \right)^{3/2} + C
\]

Let me know if you would like further details or clarifications!

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Here are five related questions to explore further:

1. How does substitution help simplify complex integrals like this one?
2. What happens if \(g(x)\) is a linear function, such as \(g(x) = x\)?
3. Can this approach be applied to integrals involving other radicals?
4. How can we check if an integral is suitable for substitution?
5. What is the geometric interpretation of solving integrals involving radicals?

**Tip:** Always verify your substitutions by differentiating your final answer to check if it matches the original integrand.

Learn how to solve the integral ∫g'(x) √(6 + 10g(x)) dx by applying the substitution method. This problem involves radicals and simplifies through substitution, with detailed steps for Grades 11-12 and college calculus students.

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