First slide

Methods of integration

Question

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

Easy

A

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

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

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

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

Solution

Let $I = \int \frac{3^{x}}{\sqrt{9^{x} - 1}} dx = \int \frac{3^{x} dx}{\sqrt{{(3^{x})}^{2} - 1}}$
put , $3^{x} = t \Rightarrow 3^{x} \log 3 dx = dt \Rightarrow 3^{x} dx = \frac{dt}{\log 3}$
$\begin{array}{l} ∴ I = \frac{1}{\log 3} \int \frac{dt}{\sqrt{t^{2} - 1}} \\ = \frac{1}{\log 3} \log [t + \sqrt{t^{2} - 1}] + C \\ = \frac{1}{\log 3} \log [3^{x} + \sqrt{9^{x} - 1}] + C \end{array}$

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