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JEE Main 202524 JanuaryEvening Shift
Exam
JEE Main
Year
2025
Date
24 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Area Under Curves

JEE Main 2025 · 24 January · Evening Shift · Mathematics · Area Under Curves

The area of the region enclosed by the curves y=e^x, y=|e^x-1| and...

The area of the region enclosed by the curves y=ex,y=∣ex−1∣y=\mathrm{e}^x, y=\left|\mathrm{e}^x-1\right| and yy-axis is:The area of the region enclosed by the curves y=e^x, y=|e^x-1| and y -axis is:

Options

  1. A1−log⁡e21-\log _{\mathrm{e}} 2
  2. Blog⁡e2\log _{\mathrm{e}} 2
  3. C1+log⁡e21+\log _e 2
  4. D2log⁡e2−12 \log _e 2-1

Correct answer

A. 1−log⁡e21-\log _{\mathrm{e}} 2

Step-by-step solution

The area of the region enclosed by the curves y=e^x, y=|e^x-1| and y -axis is:
ex=1−ex⇒2ex=1⇒ex=12⇒x=ln⁡12∫ln⁡(1/2)0[ex−(1−ex)]dx=∫ln⁡20(2ex−1)dx=2ex−x∣−ln⁡20=2−(1+ln⁡2)=1−log⁡e2\begin{aligned} & e^x=1-e^x \Rightarrow 2 e^x=1 \\ & \Rightarrow e^x=\frac{1}{2} \\ & \Rightarrow x=\ln \frac{1}{2} \\ & \int_{\ln (1 / 2)}^0\left[e^x-\left(1-e^x\right)\right] d x \\ & =\int_{\ln 2}^0\left(2 e^x-1\right) d x=2 e^x-\left.x\right|_{-\ln 2} ^0 \\ & =2-(1+\ln 2) \\ & =1-\log _e 2\end{aligned}
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