Mark the correct alternative in each of the following:
The primitive of the function $f(x)=\left(1-\frac{1}{x^{2}}\right) a^{x+\frac{1}{x}}, a>0^{\text {is }}$
A. $\frac{a^{x+\frac{1}{x}}}{\log _{e} a}$
B.$\log _{e} a \cdot a^{x+\frac{1}{x}}$
C. $\frac{a^{x+\frac{1}{x}}}{x} \log _{e} a$
D. $x \frac{a^{x+\frac{1}{x}}}{\log _{e} a}$
$\mathrm{I}=\int\left(1-\frac{1}{x^{2}}\right) a^{x+\frac{1}{x}} \mathrm{~d}_{x}$
$\Rightarrow \operatorname{let} x+\frac{1}{x}=t$
$1-\frac{1}{x^{2}}=\frac{d t}{d x}$
$=\int \mathrm{a}^{\mathrm{t}} \mathrm{dt}$
$\Rightarrow I=\frac{a^{t}}{\log _{e} a}\left(\right.$ put $\left.t=x+\frac{1}{x}\right)$
$\Rightarrow I=\frac{a^{x+\frac{1}{x}}}{\log _{e} a}+C$
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