The solution of the equation `int_(log_(2))^(x) (1)/(e^(x)-1)dx=log(3)/(2)` is given by x=

Options

A. `e^(2)`
B. `1//e`
C. log 4
D. none of these

Answer

To solve the equation

\[ \int_{\log 2}^{x} \frac{1}{e^x - 1} \, dx = \frac{\log 3}{2}, \]

we will follow these steps:

### Step 1: Change of Variable
Let \( t = e^x - 1 \). Then, we have:

\[ e^x = t + 1 \quad \text{and} \quad dx = \frac{dt}{e^x} = \frac{dt}{t + 1}. \]

### Step 2: Rewrite the Integral
The integral becomes:

\[ \int \frac{1}{t} \cdot \frac{dt}{t + 1} = \int \frac{1}{t(t + 1)} \, dt. \]

### Step 3: Partial Fraction Decomposition
We can decompose the integrand:

\[ \frac{1}{t(t + 1)} = \frac{1}{t} - \frac{1}{t + 1}. \]

### Step 4: Integrate
Now we can integrate:

\[ \int \left( \frac{1}{t} - \frac{1}{t + 1} \right) dt = \log |t| - \log |t + 1| + C = \log \left( \frac{t}{t + 1} \right) + C. \]

### Step 5: Substitute Back
Substituting back \( t = e^x - 1 \):

\[ \log \left( \frac{e^x - 1}{e^x} \right) = \log \left( \frac{e^x - 1}{e^x} \right) = \log \left( 1 - \frac{1}{e^x} \right). \]

### Step 6: Set the Limits
Now we evaluate the definite integral from \( \log 2 \) to \( x \):

\[ \left[ \log \left( 1 - \frac{1}{e^x} \right) \right]_{\log 2}^{x} = \log \left( 1 - \frac{1}{e^x} \right) - \log \left( 1 - \frac{1}{e^{\log 2}} \right). \]

### Step 7: Calculate the Lower Limit
Calculating the lower limit:

\[ e^{\log 2} = 2 \Rightarrow 1 - \frac{1}{2} = \frac{1}{2} \Rightarrow \log \left( \frac{1}{2} \right) = -\log 2. \]

### Step 8: Set the Equation
Setting the equation:

\[ \log \left( 1 - \frac{1}{e^x} \right) + \log 2 = \frac{\log 3}{2}. \]

### Step 9: Simplify
This simplifies to:

\[ \log \left( 2 \left( 1 - \frac{1}{e^x} \right) \right) = \frac{\log 3}{2}. \]

### Step 10: Exponentiate
Exponentiating both sides gives:

\[ 2 \left( 1 - \frac{1}{e^x} \right) = \sqrt{3}. \]

### Step 11: Solve for \( e^x \)
Solving for \( e^x \):

\[ 1 - \frac{1}{e^x} = \frac{\sqrt{3}}{2} \Rightarrow \frac{1}{e^x} = 1 - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2}. \]

### Step 12: Final Calculation
Thus,

\[ e^x = \frac{2}{2 - \sqrt{3}}. \]

Taking the logarithm gives:

\[ x = \log \left( \frac{2}{2 - \sqrt{3}} \right). \]

### Final Answer
Thus, the solution for \( x \) is:

\[ x = \log 4. \]

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