Evaluate : `int_(0) ^(1)(e^(x))/(sqrt(e^x-1))dx`

Answer

To evaluate the integral

\[ I = \int_{0}^{1} \frac{e^x}{\sqrt{e^x - 1}} \, dx, \]

we will use a substitution method.

### Step 1: Substitution
Let us set

\[ e^x - 1 = t^2. \]

Then, differentiating both sides gives:

\[ e^x \, dx = 2t \, dt. \]

### Step 2: Change of Limits
Now we need to change the limits of integration.

- When \( x = 0 \):
\[ e^0 - 1 = 0 \implies t^2 = 0 \implies t = 0. \]

- When \( x = 1 \):
\[ e^1 - 1 = e - 1 \implies t^2 = e - 1 \implies t = \sqrt{e - 1}. \]

### Step 3: Rewrite the Integral
Now we can rewrite the integral \( I \):

\[ I = \int_{0}^{\sqrt{e - 1}} \frac{e^x}{\sqrt{t^2}} \cdot \frac{2t}{e^x} \, dt. \]

Since \( e^x = t^2 + 1 \), we can simplify this:

\[ I = \int_{0}^{\sqrt{e - 1}} \frac{2t}{t} \, dt = \int_{0}^{\sqrt{e - 1}} 2 \, dt. \]

### Step 4: Evaluate the Integral
Now, we can evaluate the integral:

\[ I = 2 \int_{0}^{\sqrt{e - 1}} dt = 2 \left[ t \right]_{0}^{\sqrt{e - 1}} = 2 \left( \sqrt{e - 1} - 0 \right) = 2\sqrt{e - 1}. \]

### Final Answer
Thus, the value of the integral is

\[ \boxed{2\sqrt{e - 1}}. \]