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In Sage 8.6 (Mac OS),
integral( (ln( 1 + sqrt(x) )) / sqrt(x) , x,1,4) gives the result 4log(3) + 2log(4/3) - 2 that evaluates n(_) to 2.96981329957600 while numerical_integral( (ln( 1 + sqrt(x) )) / sqrt(x) ,1,4) gives the evaluation (1.8190850097688764, 2.0195900615958844e-14).

By hand, the correct result is 6n(3) - 4ln(2) - 2

In Sage 8.6 (Mac OS),
OS):

sage: integral(ln(1 + sqrt(x) )) / sqrt(x) , x,1,4) 
gives the result 4log(3) sqrt(x))/sqrt(x), x, 1, 4)
4*log(3) + 2log(4/3) 2*log(4/3) - 2 that evaluates n(_) to 2.96981329957600

sage: numerical_integral(ln(1 + sqrt(x) )) / sqrt(x) ,1,4)
gives the evaluation sqrt(x))/sqrt(x), 1, 4)
(1.8190850097688764, 2.0195900615958844e-14).
2.0195900615958844e-14).

By hand, the correct result is 6n(3) 6*ln(3) - 4ln(2) 4*ln(2) - 2.

In Sage 8.6 (Mac OS):

sage: integral(ln(1 + sqrt(x))/sqrt(x), x, 1, 4)
4*log(3) + 2*log(4/3) - 2 that evaluates n(_) to 2
sage: n(_)
2.96981329957600

while

sage: numerical_integral(ln(1 + sqrt(x))/sqrt(x), 1, 4)
(1.8190850097688764, 2.0195900615958844e-14).

By hand, the correct result is 6*ln(3) - 4*ln(2) - 2.