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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2018 · Mathematics · Indefinite Integration

What is ∫ dx 2^ x -1 equal to?

What is ∫dx2x−1\int \frac{\mathrm{dx}}{2^{\mathrm{x}}-1} equal to?

Options

  1. A\ln (2x−1)+c\left(2^{\mathrm{x}}-1\right)+\mathrm{c}
  2. Bln⁡(1−2−x)ln⁡2+c\frac{\ln \left(1-2^{-x}\right)}{\ln 2}+c
  3. Cln⁡(2−x−1)2ln⁡2+c\frac{\ln \left(2^{-\mathrm{x}}-1\right)}{2 \ln 2}+\mathrm{c}
  4. Dln⁡(1+2−x)ln⁡2+c\frac{\ln \left(1+2^{-\mathrm{x}}\right)}{\ln 2}+\mathrm{c}

Correct answer

B. ln⁡(1−2−x)ln⁡2+c\frac{\ln \left(1-2^{-x}\right)}{\ln 2}+c

Step-by-step solution

∫dx2x−1=∫dx12−x=∫2−x1−2−x⋅dx\int \frac{d x}{2^x-1}=\int \frac{d x}{\frac{1}{2^{-x}}}=\int \frac{2^{-x}}{1-2^{-x}} \cdot d x Let 1−2−x=t⇒2−x⋅log⁡2=dtdx1-2^{-\mathrm{x}}=\mathrm{t} \Rightarrow 2^{-\mathrm{x}} \cdot \log 2=\frac{\mathrm{dt}}{\mathrm{dx}}⇒2−x=1log⁡2⋅dtdx⇒2−x⋅dx=dtlog⁡2∴∫2−x1−2−xdx=1log⁡2∫dtt=1log⁡2(log⁡t)+c=1log⁡2(log⁡(1−2−x))+c\begin{aligned} & \Rightarrow 2^{-\mathrm{x}}=\frac{1}{\log 2} \cdot \frac{\mathrm{dt}}{\mathrm{dx}} \Rightarrow 2^{-\mathrm{x}} \cdot \mathrm{dx}=\frac{\mathrm{dt}}{\log 2} \\ & \therefore \int \frac{2^{-\mathrm{x}}}{1-2^{-\mathrm{x}}} \mathrm{dx}=\frac{1}{\log 2} \int \frac{\mathrm{dt}}{\mathrm{t}}=\frac{1}{\log 2}(\log \mathrm{t})+\mathrm{c} \\ & =\frac{1}{\log 2}\left(\log \left(1-2^{-\mathrm{x}}\right)\right)+\mathrm{c} \end{aligned}$$
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