\( \int _{}^{}\frac{\sin (\ln (\sqrt {1+x^{-\frac{4}{5}}}))}{(1+x^{-\frac{4}{5}})(x+x^{\frac{9}{5}})} d x \)

Calculating the integral \( I = \int \frac{\sin(\ln(\sqrt{1+x^{-\frac{4}{5}}}))}{(1+x^{-\frac{4}{5}})(x+x^{\frac{9}{5}})} \, dx \) may seem daunting at first, but we can simplify the expression step by step. First, simplify \( \ln(\sqrt{1+x^{-\frac{4}{5}}}) \): \[ \ln(\sqrt{1+x^{-\frac{4}{5}}}) = \frac{1}{2} \ln(1+x^{-\frac{4}{5}}). \] Now let's rewrite the integrand using this simplification: \[ I = \int \frac{\sin\left(\frac{1}{2} \ln(1+x^{-\frac{4}{5}})\right)}{(1+x^{-\frac{4}{5}})(x+x^{\frac{9}{5}})} \, dx. \] The argument of the sine function, \( \frac{1}{2} \ln(1+x^{-\frac{4}{5}}) \), invites us to think about the nature of \( 1+x^{-\frac{4}{5}} \). This expression approaches 1 as \( x \to \infty \). Now, notice that as \( x \to \infty \), \( x^{-\frac{4}{5}} \to 0 \), and thus: \[ \ln(1 + x^{-\frac{4}{5}}) \sim x^{-\frac{4}{5}} \text{ (for large } x). \] Therefore, the sine term behaves like: \[ \sin\left(\frac{1}{2} \ln(1+x^{-\frac{4}{5}})\right) \sim \sin\left(\frac{1}{2} x^{-\frac{4}{5}}\right). \] With \( x+x^{\frac{9}{5}} \sim x^{\frac{9}{5}} \) for large \( x \), we can see that the problem becomes manageable. However, solving this integral explicitly requires advanced techniques such as series expansions or substitution methods. It’s often beneficial to reference integral evaluation tables or specific techniques in analysis or Fourier series that can deal with such oscillatory integrands. For a complete solution, consulting material focused on integrals of logarithm and sine functions in applied physics or engineering example integrals might also provide alternative approaches and results.