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The definite integral $\int_0^{\ln 2} e^u \sqrt{1+e^u} du$ evaluates to the exact value $2\sqrt{3} - \frac{4\sqrt{2}}{3}$. This result is obtained by applying the substitution $x = 1 + e^u$, which transforms the bounds to 2 and 3, and then evaluating the antiderivative $\frac{2}{3}x^{3/2}$ at these new limits.

Conditions: The integral is $\int_0^{\ln 2} e^u \sqrt{1+e^u} du$.; The substitution $x = 1 + e^u$ is used.; The bounds are converted to 2 and 3.; The antiderivative is $\frac{2}{3}x^{3/2}$.

The definite integral $\int_0^{\ln 2} e^u \sqrt{1+e^u} du$ evaluates to the exact value $2\sqrt{3} - \frac{4\sqrt{2}}{3}$. This result is obtained by applying the substitution $x = 1 + e^u$, which transforms the bounds to 2 and 3, and then evaluating the antiderivative $\frac{2}{3}x^{3/2}$ at these new limits.

Conditions: The integral is $\int_0^{\ln 2} e^u \sqrt{1+e^u} du$.; The substitution $x = 1 + e^u$ is used.; The bounds are converted to 2 and 3.; The antiderivative is $\frac{2}{3}x^{3/2}$.