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The derivative of $1 + e^u$ is $e^u$, which appears as a factor in the integrand $e^u \sqrt{1+e^u}$. This allows the entire term $e^u du$ to be replaced by $dx$, simplifying the integral to $\int \sqrt{x} dx$.

Conditions: The integrand is $e^u \sqrt{1+e^u}$.; The substitution is $x = 1 + e^u$.; The derivative $dx/du = e^u$ is present in the integrand.

The derivative of $1 + e^u$ is $e^u$, which appears as a factor in the integrand $e^u \sqrt{1+e^u}$. This allows the entire term $e^u du$ to be replaced by $dx$, simplifying the integral to $\int \sqrt{x} dx$.

Conditions: The integrand is $e^u \sqrt{1+e^u}$.; The substitution is $x = 1 + e^u$.; The derivative $dx/du = e^u$ is present in the integrand.