Search any question & find its solution
Question:
Answered & Verified by Expert
$\int \frac{3^x d x}{\sqrt{9^x-1}}$ is equal to
Options:
Solution:
1312 Upvotes
Verified Answer
The correct answer is:
$\frac{1}{\log 3} \log \left|3^x+\sqrt{9^x-1}\right|+c$
Let $I=\int \frac{3^x d x}{\sqrt{9^x-1}}=\int \frac{3^x}{\sqrt{3^{2 x}-1}} d x$
Let $3^x=z \Rightarrow 3^x \log 3 d x=d z$
$$
I=\frac{1}{\log 3} \int \frac{z}{\sqrt{z^2-1}}
$$
$\begin{aligned} & =\frac{1}{\log 3} \log \left\{z+\sqrt{z^2-1}\right\}+c \\ & =\frac{1}{\log 3} \log \left[3^x+\sqrt{9^x-1}\right]+c\end{aligned}$
Let $3^x=z \Rightarrow 3^x \log 3 d x=d z$
$$
I=\frac{1}{\log 3} \int \frac{z}{\sqrt{z^2-1}}
$$
$\begin{aligned} & =\frac{1}{\log 3} \log \left\{z+\sqrt{z^2-1}\right\}+c \\ & =\frac{1}{\log 3} \log \left[3^x+\sqrt{9^x-1}\right]+c\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.