Choose the correct answer of the given question
`int e^(sqrtx) dx = ___ +c`

Options

A. `2 e^(sqrtx) (sqrtx-1)`
B. ` e^(sqrtx) (sqrtx-1)`
C. `2 e^(sqrtx) (sqrtx+1)`
D. `2 e^(sqrtx) (x-1)`

Answer

To solve the integral \( \int e^{\sqrt{x}} \, dx \), we will use the method of substitution. Here are the steps:

### Step 1: Substitution
Let \( z = \sqrt{x} \). Then, we have:
\[ x = z^2 \]
Now, we differentiate both sides to find \( dx \):
\[ dx = 2z \, dz \]

### Step 2: Rewrite the Integral
Substituting \( z \) and \( dx \) into the integral, we get:
\[ \int e^{\sqrt{x}} \, dx = \int e^z \cdot (2z \, dz) = 2 \int z e^z \, dz \]

### Step 3: Integration by Parts
Now we will apply integration by parts. Let:
- \( u = z \) (thus \( du = dz \))
- \( dv = e^z \, dz \) (thus \( v = e^z \))

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we have:
\[ \int z e^z \, dz = z e^z - \int e^z \, dz \]

### Step 4: Solve the Integral
Now we compute the integral:
\[ \int e^z \, dz = e^z \]
Thus, substituting back, we get:
\[ \int z e^z \, dz = z e^z - e^z = e^z (z - 1) \]

### Step 5: Substitute Back
Now substituting back for \( z \):
\[ 2 \int z e^z \, dz = 2 e^z (z - 1) = 2 e^{\sqrt{x}} \left( \sqrt{x} - 1 \right) \]

### Step 6: Add the Constant of Integration
Finally, we add the constant of integration \( C \):
\[ \int e^{\sqrt{x}} \, dx = 2 e^{\sqrt{x}} \left( \sqrt{x} - 1 \right) + C \]

Thus, the final answer is:
\[ \int e^{\sqrt{x}} \, dx = 2 e^{\sqrt{x}} \left( \sqrt{x} - 1 \right) + C \]

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