If `int(e^(x)(2-x^(2)))/((1-x)sqrt(1-x^(2)))dx=mu e^(x)((1+x)/(1-x))^(lambda)+C`, then `2(lambda+mu)` is equal to ..... .

Answer

To solve the integral

\[ I = \int \frac{e^x (2 - x^2)}{(1 - x) \sqrt{1 - x^2}} \, dx, \]

we will simplify the integrand and find the constants \(\mu\) and \(\lambda\) as described in the problem.

### Step 1: Simplify the Integrand
We can rewrite \(2 - x^2\) as \(1 - x^2 + 1\). Thus, we have:

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

### Step 2: Factor the Integrand
We can factor \(1 - x^2\) as \((1 - x)(1 + x)\):

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

This simplifies to:

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

### Step 3: Use Substitution
Let \(t = \frac{1 + x}{\sqrt{1 - x^2}}\). Then, we differentiate \(t\) with respect to \(x\):

\[ dt = \left( \frac{(1 - x^2) \cdot 1 - (1 + x)(-x)}{(1 - x^2)^{3/2}} \right) dx. \]

This leads to:

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

### Step 4: Rewrite the Integral
Now we can rewrite the integral \(I\) in terms of \(t\):

\[ I = \int e^x t \, dt. \]

### Step 5: Integrate
The integral can be computed as:

\[ I = e^x t + C = e^x \frac{1 + x}{1 - x} + C. \]

### Step 6: Compare with Given Form
We compare this with the given form:

\[ I = \mu e^x \left( \frac{1 + x}{1 - x} \right)^\lambda + C. \]

From comparison, we can see that \(\mu = 1\) and \(\lambda = \frac{1}{2}\).

### Step 7: Calculate \(2(\lambda + \mu)\)
Now we calculate:

\[ 2(\lambda + \mu) = 2\left(\frac{1}{2} + 1\right) = 2 \cdot \frac{3}{2} = 3. \]

Thus, the final answer is:

\[ \boxed{3}. \]