Q114 / 160 · TS EAMCET 2018 (05 May Shift 1) · Mathematics

$\int \frac{3^x}{\sqrt{9^x-1}} d x=$

  • Correct answer
    A
    $\frac{1}{\log 3} \log \left|3^x+\sqrt{9^x-1}\right|+c$
  • B
    $\frac{1}{\log 3} \log \left|3^x-\sqrt{9^x-1}\right|+c$
  • C
    $\frac{1}{\log 9} \log \left|3^x-\sqrt{9^x-1}\right|+c$
  • D
    $\frac{1}{\log 9} \log \left|9^x-\sqrt{9^x-1}\right|+c$

Solution

We have, $\int \frac{3^x}{\sqrt{9^x-1}} d x$ Put $3^x=t \Rightarrow 3^x \log 3 d x=d t$ $$ \begin{aligned} & \therefore \frac{1}{\log 3} \int \frac{d t}{\sqrt{t^2-1}}=\frac{1}{\log 3} \log \left|t+\sqrt{t^2-1}\right|+c \\ & \text { Put } t=3^x=\frac{1}{\log 3} \log \left|3^x+\sqrt{9^x-1}\right|+c \end{aligned} $$

Hence, the correct answer is option: 'A'.

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