Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 41

You need to use the substitution 1+x = t , such that:
1+x = t => dx = dt
Changing the variable yields:
int x*ln(1+x) dx = int (t-1)*ln t dt = int t*ln t dt - int ln t dt
You need to use the integration by parts for int t*ln t dt such that:
int udv = uv - int vdu
u = ln t => du = 1/t
dv = t=> v = t^2/2
int t*ln t dt = ( t^2/2)*ln t - int (1/t)*(t^2/2) dt
int t*ln t dt = ( t^2/2)*ln t - (1/2) int t dt
int t*ln t dt = (t^2/2)*ln t - (t^2/4) + c
You need to use the integration by parts for int ln t dt such that:
u =ln t=> du = 1/t
dv = 1=>v = t
int ln t dt = t*ln t - int t*1/t dt
int ln t dt = t*ln t - t + c
int (t-1)*ln t dt = (t^2/2)*ln t - (t^2/4) - t*ln t+ t + c
Replacing back the variable, yields:
int x*ln(1+x) dx = ((1+x)^2)/2*ln(1+x) - ((1+x)^2)/4 -(1+x)*ln (1+x) +(1+x) + c
Hence, evaluating the integral, using integration by parts, yields int x*ln(1+x) dx = ((1+x)^2)/2*ln(1+x) - ((1+x)^2)/4 -(1+x)*ln (1+x) +(1+x) + c.