Ask for solutions to two improper integrals whose integrands contain the integer powers of the sine function

Question

Let $k\in\mathbb{N}$, $\alpha\in\mathbb{R}$, and $a,b\in\mathbb{R}$ such that $a,b>0$. Then \begin{equation}\label{alpha-n=1-k-odd}\tag{q1} \int_0^\infty\frac{\sin^{2k-1}t}{t}\sin(\alpha t)\operatorname{d} t =\frac{(-1)^{k-1}}{2^{2k}}\sum_{q=0}^{2k-1}(-1)^q\binom{2k-1}{q}\ln\biggl|\frac{2q-2k+1-\alpha}{2q-2k+1+\alpha}\biggr| \end{equation} and \begin{equation}\label{a-b-n=1-k-odd}\tag{q2} \int_0^\infty\frac{\sin^{2k}(a x)}{x}\cos(bx)\operatorname{d} x =\frac{(-1)^{k-1}}{2^{2k+1}}\sum_{q=0}^{2k-1}(-1)^q\binom{2k-1}{q}\ln\biggl|\frac{4a^2(q-k)^2-b^2}{4a^2(q-k+1)^2-b^2}\biggr|. \end{equation} Recently, I derived the formulas \eqref{alpha-n=1-k-odd} and \eqref{a-b-n=1-k-odd}. The motivation comes from related results on pp. 464--471 in the monogrpah [1] below and those results in the papers [2] to [6].

Let $m,n\in\mathbb{N}$ such that $n\ge2$ and $m-n\ge1$ is odd and let $a,b,\alpha\in\mathbb{R}$ such that $a>0$ and $b\ge0$. How to compute or where to find the improper integrals \begin{equation*} \int_0^\infty\frac{\sin^{m-1}t}{t^{n}}\sin(\alpha t)\operatorname{d} t \quad\text{and}\quad \int_0^\infty\frac{\sin^{m}(a x)}{x^{n}}\cos(bx)\operatorname{d} x? \end{equation*} Due to related results in the monographs [7] and [8], I think that this problem is interesting and significant.

References

1. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/Academic Press, Amsterdam, 2015; available online at https://doi.org/10.1016/B978-0-12-384933-5.00013-8.
2. Q.-M. Luo, B.-N. Guo, and F. Qi, Evaluation of a class of improper integrals of the first kind, Math. Gaz. 87 (2003), no. 510, 534--539; available online at https://doi.org/10.1017/S0025557200173863.
3. Q.-M. Luo and F. Qi, Evaluation of a class of the first kind improper integrals $\int_0^\infty\bigl(\frac{\sin(\beta x)}{x}\bigr)^n\cos(bx)\operatorname{d} x$, Octogon Math. Mag. 11 (2003), no. 1, 76--81.
4. Q.-M. Luo and F. Qi, Evaluation of the improper integrals ${\int_0^\infty\frac{\sin^{2m}(a x)}{x^{2n}}\cos(bx)\operatorname{d} x}$ and ${\int_0^\infty\frac{\sin^{2m+1}(a x)}{x^{2n+1}}\cos(bx)\operatorname{d} x}$, Austral. Math. Soc. Gaz. 30 (2003), no. 2, 86--89.
5. F. Qi, Q.-M. Luo, and B.-N. Guo, Evaluations of the improper integrals $\int_0^\infty\frac{\sin^{2m}(\alpha x)}{x^{2n}}\operatorname{d} x$ and $\int_0^\infty\frac{\sin^{2m+1}(\alpha x)}{x^{2n+1}}\operatorname{d} x$, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 11 (2004), no. 3, 189--196.
6. A. Sofo, On the integral $\frac{(\sin x)^{p+q}}{x^q}$, Internat. J. Math. Ed. Sci. Tech. 29 (1998), no. 6, 914--918.
7. C. I. Valean, (Almost) Impossible Integrals, Sums, and Series, with a foreword by Paul J. Nahin, Problem Books in Mathematics, Springer, Cham, 2019; available online at https://doi.org/10.1007/978-3-030-02462-8.
8. C. I. Valean, More (almost) Impossible Integrals, Sums, and Series---A New Collection of Friendish Problems and Surprising Solutions, Problem Books in Mathematics, Springer, Cham, 2023; available online at https://doi.org/10.1007/978-3-031-21262-8.