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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Definite Integration

NDA 2022 · Mathematics · Definite Integration

What is ∫_0^1 ((1)/(x)-1) d x equal to?

What is ∫01ln⁡(1x−1)dx\int_0^1 \ln \left(\frac{1}{x}-1\right) d x equal to?

Options

  1. A−1-1
  2. B00
  3. C11
  4. Dln⁡2\ln 2

Correct answer

B. 00

Step-by-step solution

I=∫01ln⁡(1x−1)dx=∫01[ln⁡(1−x)−ln⁡x]dx=∫01ln⁡(1−x)dx−∫01ln⁡xdx=∫01ln⁡(1−x)dx−∫01ln⁡(1−x)dx=0\begin{aligned} & I=\int_0^1 \ln \left(\frac{1}{x}-1\right) d x=\int_0^1[\ln (1-x)-\ln x] d x \\ & =\int_0^1 \ln (1-x) d x-\int_0^1 \ln x d x \\ & =\int_0^1 \ln (1-x) d x-\int_0^1 \ln (1-x) d x=0\end{aligned}
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