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 will use substitution. Here’s the step-by-step solution:

### Step 1: Choose a substitution
Let \( t = \sqrt{x + 1} \). Then, squaring both sides gives:
\[ x + 1 = t^2 \implies x = t^2 - 1 \]

### Step 2: Differentiate to find \( dx \)
Differentiating both sides with respect to \( t \):
\[ dx = 2t \, dt \]

### Step 3: Substitute in the integral
Now substitute \( x \) and \( dx \) in the integral:
\[ I = \int \frac{t^2 - 1}{t} \cdot 2t \, dt \]
This simplifies to:
\[ I = \int 2(t^2 - 1) \, dt \]

### Step 4: Simplify the integral
Now distribute the 2:
\[ I = 2 \int (t^2 - 1) \, dt = 2 \left( \int t^2 \, dt - \int 1 \, dt \right) \]

### Step 5: Integrate
Now we can integrate each term:
\[ \int t^2 \, dt = \frac{t^3}{3} \quad \text{and} \quad \int 1 \, dt = t \]
Thus,
\[ I = 2 \left( \frac{t^3}{3} - t \right) + C \]

### Step 6: Substitute back for \( t \)
Now substitute back \( t = \sqrt{x + 1} \):
\[ I = 2 \left( \frac{(\sqrt{x + 1})^3}{3} - \sqrt{x + 1} \right) + C \]
This simplifies to:
\[ I = \frac{2}{3} (x + 1)^{3/2} - 2\sqrt{x + 1} + C \]

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

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