`intsqrt(e^(x))dx=?`

Options

A. `sqrt(e^(x))+C`
B. `2sqrt(e^(x))+C`
C. `(1)/(2)sqrt(e^(x))+C`
D. none of these

Explanation

To solve the integral \(\int \sqrt{e^x} \, dx\), we can follow these steps:

### Step 1: Rewrite the Integral
First, we rewrite the integral in a more manageable form:
\[ \int \sqrt{e^x} \, dx = \int e^{x/2} \, dx \]

### Step 2: Integrate
Now, we can integrate \(e^{x/2}\). The integral of \(e^{kx}\) is \(\frac{1}{k} e^{kx}\). Here, \(k = \frac{1}{2}\):
\[ \int e^{x/2} \, dx = \frac{1}{\frac{1}{2}} e^{x/2} + C = 2 e^{x/2} + C \]

### Step 3: Rewrite the Result
Finally, we can rewrite the result in terms of the square root:
\[ 2 e^{x/2} + C = 2 \sqrt{e^x} + C \]

### Final Answer
Thus, the integral \(\int \sqrt{e^x} \, dx\) is:
\[ \int \sqrt{e^x} \, dx = 2 \sqrt{e^x} + C \]

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