I looked up the information at wikipedia which is providing for the substitution method.May i ask some additional help to get it right for the following specific integral using the substitution method?
if $\,y(t) = sin(ωt+\theta), \,where \,\,T_0=\frac{2π}{ω_ο} $ How do i convert from $\int_0^{T_0}y(t)\,dt$ to $\int_0^{2π}sin(ωt+\theta)\,d(ωt)$?
I can understand the first practical example wikipedia is providing for the cos(u) integral but i cannot correctly implement it in the above problem. Please, if you answer it be detailed.I do not feel experienced enough with the substitution method.
Note that
$$\int_0^{T_0}y(t)\,dt=\int_0^{T_0}sin(ωt+\theta)\,dt$$
Let
$$s=\omega t \implies ds=d\omega t$$
thus
$$\int_0^{T_0}y(t)\,dt=\int_0^{\omega T_0}sin(s+\theta)\,ds =\int_0^{2\pi}sin(s+\theta)\,ds =\int_0^{2\pi}sin(ωt+\theta)\,d(\omega t) $$