PartialIntegrate[exp, ap, t]
does a partial integration of the definite integral Integrate[exp,{t,0,1}]
, with ap
the factor that is to be integrated and exp/ap
the factor that is to be differentiated.
Overview, IntegrateByParts, Integrate2.
[f[x] g[x], g[x], {x, 0, 1}] PartialIntegrate
f[x_] = Integrate[Log[3 x + 2], x]
g[x_] = D[1/Log[3 x + 2], x]
Integrate[PartialIntegrate[f[x] g[x], f[x], x], {x, 0, 1}] // FullSimplify
Integrate[f[x] g[x], {x, 0, 1}] // Simplify
Clear[f, g]