$\int \frac{3^{\mathrm{x}}}{\sqrt{9^{\mathrm{x}}-1}}\mathrm{dx}$ is equal to

Solution:

Let $\mathrm{I}=\int \frac{3^{\mathrm{x}}}{\sqrt{9^{\mathrm{x}}-1}}\mathrm{dx}=\int \frac{3^{\mathrm{x}}\mathrm{dx}}{\sqrt{{\left(3^{\mathrm{x}}\right)}^2-1}}$

put ,$3^{\mathrm{x}}=\mathrm{t}\Rightarrow 3^{\mathrm{x}}\log 3\mathrm{dx}=\mathrm{dt}\Rightarrow 3^{\mathrm{x}}\mathrm{dx}=\frac{\mathrm{dt}}{\log 3}$

$\begin{array}{l}\therefore \mathrm{I}=\frac{1}{\log 3}\int \frac{\mathrm{dt}}{\sqrt{{\mathrm{t}}^2-1}}\\ =\frac{1}{\log 3}\log \left[\mathrm{t}+\sqrt{{\mathrm{t}}^2-1}\right]+\mathrm{C}\\ =\frac{1}{\log 3}\log \left[3^{\mathrm{x}}+\sqrt{9^{\mathrm{x}}-1}\right]+\mathrm{C}\end{array}$