Question

The area enclosed between the curve \( y = \log_e(x + e) \) and the coordinate axes is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For finding the area under curves involving logarithms, use substitution and evaluate the integral carefully, keeping in mind the behavior of the logarithmic function at both the upper and lower limits.

Answer 1

The Correct Option is A

 

Required area \[ A = \int_{1 - e}^{0} y \, dx = \int_{1 - e}^{0} \log_e (x + e) \, dx \] Put \( x + e = t \Rightarrow dx = dt \), also when \( x = 1 - e \), \( t = 1 \) and when \( x = 0 \), \( t = e \). \[ \therefore A = \int_{1}^{e} \log_e t \, dt = \left[ t \log_e t - t \right]_{1}^{e} \] \[ e - e - 0 + 1 = 1 \]