Mark the correct alternative in each of the following:

Question:

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. $x \frac{a^{x+\frac{1}{x}}}{\log _{e} a}$

Solution:

$\mathrm{I}=\int\left(1-\frac{1}{x^{2}}\right) a^{x+\frac{1}{x}} \mathrm{~d}_{x}$

$\Rightarrow \operatorname{let} x+\frac{1}{x}=t$

$1-\frac{1}{x^{2}}=\frac{d t}{d x}$

$=\int \mathrm{a}^{\mathrm{t}} \mathrm{dt}$

$\Rightarrow I=\frac{a^{t}}{\log _{e} a}\left(\right.$ put $\left.t=x+\frac{1}{x}\right)$

$\Rightarrow I=\frac{a^{x+\frac{1}{x}}}{\log _{e} a}+C$

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