Question

$(1+\frac{1}{x^2})a^{x-\frac{1}{x}}$

Answer 1

$\frac{a^{x-\frac{1}{x}}}{\ln a} + C$

Answer 2

The problem requires finding the indefinite integral of the given expression. We use the substitution method. Let $u = x - \frac{1}{x}$. Differentiating $u$ with respect to $x$ gives $du = (1 + \frac{1}{x^2}) dx$. The integral transforms into $\int a^u du$, which evaluates to $\frac{a^u}{\ln a} + C$. Substituting back $u = x - \frac{1}{x}$ yields the final result.