I am working with General Forced Response for vibrations and I am trying to establish an easy procedure for solving integrals such as the one below.
$$ \int_{0}^{t}\tau\, \sin\left(\,{\omega_{n}\left[\,{t - \tau}\,\right]}\,\right)\,\mathrm{d}\tau $$
A quick conclusion is that first utilizing u-substitution, such that $u = (t - τ)$, is a good place to start to make things simpler with partial integration and such later. However, upon doing so, the result proposed by Symbolab has me a bit perplexed: $$\int_t^0 -(-u+t)sin((ω_n(u))\;du$$ I am currently trying to understand where the constant negative sign $(-)$ before $(-u + t)$ comes from, since, surely, $(t-u)=τ\;$ as per the initial u-substitution? Does $(-u+t)$ then somehow not satisfy the original expression of $τ\;$ on its own?
I suspect that it may involve the new positioning of the internals of the parenthesis but I would very much like to hear from others on this. Thanks in advance!
It comes from the differential: if $u = t-\tau$, then $du = -d\tau$, so $d\tau = -du$.