Question

Find the value of the integral: \[ \int_0^e \log_e x \, dx \]

• A. \( 1 \)
• B. \( e - 1 \)
• C. \( \frac{e}{2} - 1 \)
• D. \( 0 \)

Hint:

For integrals involving logarithms, use integration by parts to simplify the expression.

Answer 1

The Correct Option is C

The integral to be evaluated is:\[I = \int_0^e \log_e x \, dx\]Integration by parts will be employed, utilizing the formula:\[\int u \, dv = uv - \int v \, du\]Setting:\[u = \log_e x, \quad dv = dx\]This yields:\[du = \frac{1}{x} dx, \quad v = x\]Applying the integration by parts formula results in:\[I = x \log_e x \bigg|_0^e - \int_0^e x \cdot \frac{1}{x} dx\]\[I = e \log_e e - 0 \cdot \log_e 0 - \int_0^e 1 \, dx\]\[I = e \cdot 1 - 0 - \left[ x \right]_0^e\]\[I = e - (e - 0) = e - e = 0\]Thus, the value of the integral is \( 0 \).