TrickIntegrate[(1 - t)^(a * Epsilon - 1) g[t], t] does an integration trick for the definite integral of ((1-t)^{a \;\text{Epsilon}-1} g[t]) from 0 to 1, yielding g[1]/a/Epsilon + Hold[Integrate][(1-t)^{a Epsilon-1} (g[t]-g[1]),{t,0,1}]
TrickIntegrate[t^(a Epsilon-1) g[t], t] gives \frac{g[0]}{a \;\text{Epsilon}}+ Hold[Integrate][t^{a \;\text{Epsilon}-1} (g[t]-g[0]),{t,0,1}], provided g[1] and g[0] exist.
TrickIntegrate[(1 - t)^(a Epsilon - 1) g[t], t]\text{Hold}[\text{Integrate}]\left[g(t) (1-t)^{a \varepsilon -1},\{t,0,1\}\right]
TrickIntegrate[t^(a Epsilon - 1) g[t], t]\text{Hold}[\text{Integrate}]\left[g(t) t^{a \varepsilon -1},\{t,0,1\}\right]