`int(1+x+sqrt(x+x^(2)))/(sqrtx+sqrt(1+x))dx=`

Options

A. `(1)/(2)sqrt(1+x)+c`
B. `(2)/(3)(1+x)^(3//2)+c`
C. `sqrt(1+x)+c`
D. `2(1+x)^(3//2)+c`

Answer

To solve the integral

\[ \int \frac{1 + x + \sqrt{x + x^2}}{\sqrt{x} + \sqrt{1 + x}} \, dx, \]

we will simplify the integrand step by step.

### Step 1: Simplify the expression inside the integral

We start by rewriting \(1 + x\) in a different form. We can express \(1 + x\) as \((\sqrt{1 + x})^2\). Thus, we have:

\[ \sqrt{x + x^2} = \sqrt{x(1 + x)}. \]

Now, substituting this back into the integral, we get:

\[ \int \frac{(\sqrt{1 + x})^2 + \sqrt{x(1 + x)}}{\sqrt{x} + \sqrt{1 + x}} \, dx. \]

### Step 2: Factor out common terms

Next, we can factor out \(\sqrt{x}\) from the numerator:

\[ \sqrt{x}(\sqrt{1 + x} + 1). \]

So the integral becomes:

\[ \int \frac{\sqrt{x}(\sqrt{1 + x} + 1)}{\sqrt{x} + \sqrt{1 + x}} \, dx. \]

### Step 3: Cancel common terms

Now, we can simplify the expression further. The numerator \(\sqrt{x}(\sqrt{1 + x} + 1)\) and the denominator \(\sqrt{x} + \sqrt{1 + x}\) share a common term. This allows us to cancel out the common terms:

\[ \int \sqrt{1 + x} \, dx. \]

### Step 4: Integrate

Now we need to integrate \(\sqrt{1 + x}\). We can use the power rule for integration:

\[ \int (1 + x)^{1/2} \, dx = \frac{(1 + x)^{3/2}}{3/2} + C = \frac{2}{3}(1 + x)^{3/2} + C. \]

### Final Solution

Thus, the final solution to the integral is:

\[ \frac{2}{3}(1 + x)^{3/2} + C. \]