# int x ln(1 + x) dx First make a substitution and then use integration by parts to evaluate the integral

## Expert Answers 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.

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