To solve the integral \(\int \frac{-10}{\left(\sqrt{x+1} + \sqrt{x}\right)} \, dx\), let's start by simplifying the integrand and applying a suitable substitution.

### Step 1: Simplify the integrand
Rewrite the expression under the integral:
\[
\int \frac{-10}{\sqrt{x+1} + \sqrt{x}} \, dx
\]

### Step 2: Substitute \( u = \sqrt{x+1} + \sqrt{x} \)
Let:
\[
u = \sqrt{x+1} + \sqrt{x}
\]
Differentiate both sides with respect to \(x\):
\[
du = \frac{1}{2\sqrt{x+1}} \cdot 1 + \frac{1}{2\sqrt{x}} \cdot 1 \, dx
\]
Simplify this:
\[
du = \frac{1}{2\sqrt{x+1}} + \frac{1}{2\sqrt{x}} \, dx
\]

Next, let's solve this integral step by step.

### Step 3: Solve the integral
The integral becomes:
\[
\int \frac{-10}{u} \, du
\]

This is a basic integral:
\[
-10 \ln |u| + C
\]
where \(C\) is the constant of integration.

### Step 4: Back-substitute \(u\)
Now, substitute back \(u = \sqrt{x+1} + \sqrt{x}\):
\[
-10 \ln |\sqrt{x+1} + \sqrt{x}| + C
\]

So the solution to the integral is:
\[
\int \frac{-10}{\sqrt{x+1} + \sqrt{x}} \, dx = -10 \ln |\sqrt{x+1} + \sqrt{x}| + C
\]

Would you like to see more details on any step or ask further questions?

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Here are 5 related questions:
1. How do you integrate functions involving square roots in the denominator?
2. What are the common substitution techniques in integral calculus?
3. How can you simplify integrals involving logarithmic functions?
4. What are the best practices for handling complex integrals?
5. How can you verify the result of an integral using differentiation?

**Tip:** When dealing with square roots in integrals, substitution often simplifies the problem significantly, so look for patterns that can reduce complexity.

Explore the step-by-step solution to integrate -10 divided by the square root of (x+1) plus the square root of x. This integral involves substitution and logarithmic functions, suitable for college-level calculus students.

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