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$\int \frac{3^x d x}{\sqrt{9^x-1}}$ is equal to
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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 $3^x=z \Rightarrow 3^x \log 3 d x=d z$
$$
I=\frac{1}{\log 3} \int \frac{z}{\sqrt{z^2-1}}
$$
$\begin{aligned} & =\frac{1}{\log 3} \log \left\{z+\sqrt{z^2-1}\right\}+c \\ & =\frac{1}{\log 3} \log \left[3^x+\sqrt{9^x-1}\right]+c\end{aligned}$
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