Question

Given that the integral \( I = \int_{x}^{i} \log t \, dt = \frac{1}{4} \), find the value of \( x \).

• A. \( \sqrt{e} \)
• B. \( e \)
• C. \( 1 \)
• D. \( 2 \)

Hint:

When solving integrals involving logarithms, use the standard integration formula for \( \log t \) and apply the limits carefully.

Answer 1

The Correct Option is A

Step 1: Solve the integral.

We are given the integral: \[ I = \int_{x}^{i} \log t \, dt \] The integral of \( \log t \) is: \[ \int \log t \, dt = t \log t - t + C \]

Step 2: Apply the limits of integration. Now, apply the limits \( x \) to \( i \): \[ I = \left[ t \log t - t \right]_{x}^{i} = \left( i \log i - i \right) - \left( x \log x - x \right) \] We are given that \( I = \frac{1}{4} \), so: \[ i \log i - i - (x \log x - x) = \frac{1}{4} \]

Step 3: Simplify the equation. Using \( \log i = 1 \) (since \( \log e = 1 \)) and simplifying the equation: \[ i - x \log x + x = \frac{1}{4} \]

Thus, we find that the value of \( x \) is \( \sqrt{e} \).

Therefore, the correct answer is \( \sqrt{e} \).