∫3x9x−1dx is equal to

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

1. A

$\frac{1}{\mathrm{log}3}\mathrm{log}\left|{3}^{\mathrm{x}}+\sqrt{{9}^{\mathrm{x}}-1}\right|+\mathrm{C}$

2. B

$\frac{1}{\mathrm{log}3}\mathrm{log}\left|{9}^{\mathrm{x}}+\sqrt{{9}^{\mathrm{x}}-1}\right|+\mathrm{C}$

3. C

$\frac{1}{\mathrm{log}9}\mathrm{log}\left|{3}^{\mathrm{x}}+\sqrt{{9}^{\mathrm{x}}-1}\right|+\mathrm{C}$

4. D

$\frac{1}{\mathrm{log}9}\mathrm{log}\left|{3}^{\mathrm{x}}-\sqrt{{9}^{\mathrm{x}}-1}\right|+\mathrm{C}$

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### 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 ,${3}^{\mathrm{x}}=\mathrm{t}⇒{3}^{\mathrm{x}}\mathrm{log}3\mathrm{dx}=\mathrm{dt}⇒{3}^{\mathrm{x}}\mathrm{dx}=\frac{\mathrm{dt}}{\mathrm{log}3}$

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