`int_(1)^(e )(1)/(x sqrt(log x))dx=`

Options

A. 2
B. `-2`
C. `(1)/(2)`
D. `-(1)/(2)`

Answer

To solve the integral \( \int_{1}^{e} \frac{1}{x \sqrt{\log x}} \, dx \), we will use a substitution method. Here are the steps:

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

### Step 2: Change the limits of integration
When \( x = 1 \):
\[ t = \log 1 = 0 \]
When \( x = e \):
\[ t = \log e = 1 \]
Thus, the limits change from \( x: 1 \to e \) to \( t: 0 \to 1 \).

### Step 3: Substitute in the integral
Now substitute \( x \) and \( dx \) in the integral:
\[ \int_{1}^{e} \frac{1}{x \sqrt{\log x}} \, dx = \int_{0}^{1} \frac{1}{e^t \sqrt{t}} \cdot e^t \, dt = \int_{0}^{1} \frac{1}{\sqrt{t}} \, dt \]

### Step 4: Evaluate the integral
The integral \( \int_{0}^{1} \frac{1}{\sqrt{t}} \, dt \) can be computed as follows:
\[ \int \frac{1}{\sqrt{t}} \, dt = 2\sqrt{t} + C \]
Now, we evaluate it from \( 0 \) to \( 1 \):
\[ \left[ 2\sqrt{t} \right]_{0}^{1} = 2\sqrt{1} - 2\sqrt{0} = 2 - 0 = 2 \]

### Final Answer
Thus, the value of the integral \( \int_{1}^{e} \frac{1}{x \sqrt{\log x}} \, dx \) is \( 2 \).

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