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.
PartialIntegrate[f[x] g[x], g[x], {x, 0, 1}]-(f(x) \int g(x) \, dx\text{/.}\, x\to 0)+(f(x) \int g(x) \, dx\text{/.}\, x\to 1)-\int_0^1 f'(x) (\int g(x) \, dx) \, dx
f[x_] = Integrate[Log[3 x + 2], x]
g[x_] = D[1/Log[3 x + 2], x]\left(x+\frac{2}{3}\right) \log (3 x+2)-x
-\frac{3}{(3 x+2) \log ^2(3 x+2)}
Integrate[PartialIntegrate[f[x] g[x], f[x], x], {x, 0, 1}] // FullSimplify-\frac{1}{\log (5)}
Integrate[f[x] g[x], {x, 0, 1}] // Simplify-\frac{1}{\log (5)}
Clear[f, g]