https://r2.erweima.ai/mathbot/image/17250809866951kvlhwu7.jpg

 Resuelve poc cambio de variable o sustitucion ,  la raiz es x+1  

To solve the integral \(\int \frac{x}{\sqrt{x + 1}} \, dx\) using a substitution method, we can follow these steps:

### Step 1: Substitution
Let's substitute \( u = x + 1 \).

Then, the differential \( du \) will be:
\[
du = dx
\]

Since \( u = x + 1 \), we also have:
\[
x = u - 1
\]

### Step 2: Rewrite the Integral
Now, substitute the values into the integral:
\[
\int \frac{x}{\sqrt{x + 1}} \, dx = \int \frac{u - 1}{\sqrt{u}} \, du
\]

### Step 3: Simplify the Expression
The integral becomes:
\[
\int \left( \frac{u}{\sqrt{u}} - \frac{1}{\sqrt{u}} \right) du = \int \left( u^{\frac{1}{2}} - u^{-\frac{1}{2}} \right) du
\]

### Step 4: Integrate
Now, integrate each term separately:
\[
\int u^{\frac{1}{2}} \, du = \frac{2}{3} u^{\frac{3}{2}}
\]
\[
\int u^{-\frac{1}{2}} \, du = 2u^{\frac{1}{2}}
\]

So, the integral becomes:
\[
\frac{2}{3} u^{\frac{3}{2}} - 2u^{\frac{1}{2}} + C
\]

### Step 5: Substitute Back
Finally, substitute back \( u = x + 1 \) into the expression:
\[
\frac{2}{3} (x + 1)^{\frac{3}{2}} - 2(x + 1)^{\frac{1}{2}} + C
\]

So the solution to the integral is:
\[
\int \frac{x}{\sqrt{x + 1}} \, dx = \frac{2}{3} (x + 1)^{\frac{3}{2}} - 2(x + 1)^{\frac{1}{2}} + C
\]

Would you like further details or explanations on any part of the solution?

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Here are five related questions to expand your understanding:

1. How would you approach the integral if it were \(\int \frac{x}{\sqrt{x + 2}} \, dx\)?
2. What happens if you substitute \( u = \sqrt{x + 1} \) instead?
3. Can you derive the general formula for \(\int \frac{x}{\sqrt{x + a}} \, dx\)?
4. How does the integration process change if the integrand includes a logarithmic function?
5. What are the common methods for integrating rational functions involving radicals?

**Tip:** When solving integrals using substitution, always remember to revert back to the original variable at the end of the integration process.

Learn how to solve the integral ∫(x/√(x + 1)) dx step-by-step using the substitution method. This problem involves integrating rational functions with square roots, suitable for advanced high school or college-level students.

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