Evaluate the following integrals:
`int frac{x}{sqrt(x+1)}dx`

Answer

To evaluate the integral \( I = \int \frac{x}{\sqrt{x+1}} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{x}{\sqrt{x+1}} \, dx \]
To simplify the integral, we can add and subtract \( \frac{1}{\sqrt{x+1}} \) in the numerator:
\[ I = \int \left( \frac{x+1}{\sqrt{x+1}} - \frac{1}{\sqrt{x+1}} \right) \, dx \]
This can be rewritten as:
\[ I = \int \frac{x+1}{\sqrt{x+1}} \, dx - \int \frac{1}{\sqrt{x+1}} \, dx \]

### Step 2: Simplify the First Integral
Now, simplify the first integral:
\[ \frac{x+1}{\sqrt{x+1}} = \frac{x}{\sqrt{x+1}} + \frac{1}{\sqrt{x+1}} = \sqrt{x+1} + \frac{1}{\sqrt{x+1}} \]
Thus, we can express \( I \) as:
\[ I = \int \sqrt{x+1} \, dx + \int \frac{1}{\sqrt{x+1}} \, dx - \int \frac{1}{\sqrt{x+1}} \, dx \]
The two \( \int \frac{1}{\sqrt{x+1}} \, dx \) terms cancel each other out, so we have:
\[ I = \int \sqrt{x+1} \, dx \]

### Step 3: Evaluate the Integral
Now we need to evaluate \( \int \sqrt{x+1} \, dx \). We can use the power rule for integration:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
In this case, we have:
\[ \int (x+1)^{1/2} \, dx = \frac{(x+1)^{3/2}}{3/2} + C = \frac{2}{3} (x+1)^{3/2} + C \]

### Step 4: Combine Results
Now, we combine our results:
\[ I = \frac{2}{3} (x+1)^{3/2} + C \]

### Step 5: Final Answer
Thus, the final result of the integral is:
\[ \int \frac{x}{\sqrt{x+1}} \, dx = \frac{2}{3} (x+1)^{3/2} - 2\sqrt{x+1} + C \]