The value of `int _(0)^(2) (3^sqrt(x))/(sqrt(x)) ` dx is

Options

A. `(2)/(log3)(3^(sqrt2)-1)`
B. 0
C. `2(sqrt2)/(log3)`
D. `(3^(sqrt2))/(sqrt2)`

Explanation

To solve the integral \( I = \int_{0}^{2} \frac{3^{\sqrt{x}}}{\sqrt{x}} \, dx \), we will use a substitution method.

### Step 1: Substitution
Let \( u = \sqrt{x} \). Then, we have:
\[ x = u^2 \quad \text{and} \quad dx = 2u \, du \]

### Step 2: Change the limits of integration
When \( x = 0 \):
\[ u = \sqrt{0} = 0 \]
When \( x = 2 \):
\[ u = \sqrt{2} \]

### Step 3: Rewrite the integral
Now, substituting \( u \) into the integral, we get:
\[ I = \int_{0}^{\sqrt{2}} \frac{3^{u}}{u} (2u) \, du \]
This simplifies to:
\[ I = 2 \int_{0}^{\sqrt{2}} 3^{u} \, du \]

### Step 4: Evaluate the integral
The integral \( \int 3^{u} \, du \) can be evaluated as follows:
\[ \int 3^{u} \, du = \frac{3^{u}}{\ln(3)} + C \]
Thus,
\[ I = 2 \left[ \frac{3^{u}}{\ln(3)} \right]_{0}^{\sqrt{2}} \]

### Step 5: Substitute the limits
Now, substituting the limits:
\[ I = 2 \left( \frac{3^{\sqrt{2}}}{\ln(3)} - \frac{3^{0}}{\ln(3)} \right) \]
\[ I = 2 \left( \frac{3^{\sqrt{2}} - 1}{\ln(3)} \right) \]

### Final Result
Thus, the value of the integral is:
\[ I = \frac{2(3^{\sqrt{2}} - 1)}{\ln(3)} \]

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