Question:
Mark the correct alternative in each of the following:
The primitive of the function $\mathrm{f}(\mathrm{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:
$\mathrm{I}=\int\left(1-\frac{1}{x^{2}}\right) a^{x+\frac{1}{x}} \mathrm{~d} \chi$
$\Rightarrow \operatorname{let} x+\frac{1}{x}=t$
$1-\frac{1}{x^{2}}=\frac{d t}{d x}$
$=\int a^{t} d t$
$\Rightarrow I=\frac{a^{t}}{\log _{e} a}\left(\right.$ put $\left.t=x+\frac{1}{x}\right)$
$\Rightarrow I=\frac{a^{x+\frac{1}{x}}}{\log _{e} a}+C$