Evaluate: `int(e^x)/(sqrt(4-e^(2x)))dx`

Options

A. 0
B. 1
C. 2
D. 4

Answer

To evaluate the integral

\[ \int \frac{e^x}{\sqrt{4 - e^{2x}}} \, dx, \]

we can follow these steps:

### Step 1: Substitution
Let \( t = e^x \). Then, differentiate both sides to find \( dx \):
\[ dt = e^x \, dx \quad \Rightarrow \quad dx = \frac{dt}{t}. \]

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we have:
\[ \int \frac{t}{\sqrt{4 - t^2}} \cdot \frac{dt}{t} = \int \frac{1}{\sqrt{4 - t^2}} \, dt. \]

### Step 3: Recognize the Integral Form
The integral \( \int \frac{1}{\sqrt{4 - t^2}} \, dt \) is a standard form, which can be integrated using the formula:
\[ \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C, \]
where \( a = 2 \) in our case.

### Step 4: Apply the Formula
Applying the formula, we get:
\[ \int \frac{1}{\sqrt{4 - t^2}} \, dt = \sin^{-1}\left(\frac{t}{2}\right) + C. \]

### Step 5: Substitute Back
Now, substitute back \( t = e^x \):
\[ \sin^{-1}\left(\frac{e^x}{2}\right) + C. \]

### Final Answer
Thus, the evaluated integral is:
\[ \int \frac{e^x}{\sqrt{4 - e^{2x}}} \, dx = \sin^{-1}\left(\frac{e^x}{2}\right) + C. \]

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