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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.

The substitution $x = 1 + e^u$ simplifies the integrand because its derivative $dx = e^u du$ exactly matches the $e^u du$ term in the original integral. This allows the complex expression $e^u \sqrt{1+e^u} du$ to be rewritten as the simpler radical $\sqrt{x} dx$.

Conditions: The integrand contains $e^u \sqrt{1+e^u}$.; The substitution is $x = 1 + e^u$.; The differential is $dx = e^u du$.

The substitution $x = 1 + e^u$ simplifies the integrand because its derivative $dx = e^u du$ exactly matches the $e^u du$ term in the original integral. This allows the complex expression $e^u \sqrt{1+e^u} du$ to be rewritten as the simpler radical $\sqrt{x} dx$.

Conditions: The integrand contains $e^u \sqrt{1+e^u}$.; The substitution is $x = 1 + e^u$.; The differential is $dx = e^u du$.