Question

The area of the region enclosed between the curve \[ y=\log_e(x+e) \] and the coordinate axes is

• A. \(4\)
• B. \(3\)
• C. \(2\)
• D. \(1\) \bigskip

Hint:

For regions bounded by a logarithmic curve and coordinate axes, first find the intercepts carefully. A substitution of the form \[ u=x+a \] usually converts the integral into the standard form \[ \int \ln u\,du. \]

Answer 1

The Correct Option is D

Concept: Area between curve \(y=\ln(x+e)\), x-axis, and y-axis. Find x-intercept, set up integral, use substitution.

Step 1: x-intercept: \(\ln(x+e)=0 \Rightarrow x+e=1 \Rightarrow x=1-e\). Region from \(x=1-e\) to \(x=0\).

Step 2: Area = \(\int_{1-e}^0 \ln(x+e)dx\). Let \(u=x+e\), \(du=dx\). Limits: \(x=1-e \Rightarrow u=1\); \(x=0 \Rightarrow u=e\). Integral = \(\int_1^e \ln u\,du = [u\ln u - u]_1^e = (e\cdot1 - e) - (1\cdot0 - 1) = 0 - (-1) = 1\).

Step 3: Write the final answer. \(\boxed{1}\)