An analytic approach to compute integrals: \(J_ n(\alpha)=\int^ \infty_ 0z^ n\bigl[Ai(z)\bigr]^ \alpha dz\), \(J_ n'(\alpha)=\int_ 0^ \infty\bigl[Ai'(z)\bigr]^ \alpha dz\) is presented, where \(Ai'[z]\) is the derivative of the Airy function \(Ai(z)\) and \(\alpha\) is any real number. Its mathematical basis lies on the introduction of an auxiliary function: \[ i_{k,\beta}(\alpha)=\int^ \infty_ 0z^ k\bigl[Ai(z)\bigr]^ \beta\bigl[Ai'(z)\bigr] ^{\alpha- \beta}dz,\quad\text{with }\beta\leq\alpha. \] and reduction to a linear partial difference equation with two variables and then derivation of recurrence relations for \(J_ n\) and \(J_ n'\).