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To evaluate the definite integral $\int_0^{\ln 2} e^u \sqrt{1+e^u} \, du$, use the substitution $x = 1 + e^u$, which yields $dx = e^u \, du$. Convert the limits of integration from $u=0$ and $u=\ln 2$ to $x=2$ and $x=3$, transforming the integral into $\int_2^3 \sqrt{x} \, dx$. Evaluating the antiderivative $\frac{2}{3}x^{3/2}$ at these new bounds gives the result $2\sqrt{3} - \frac{4\sqrt{2}}{3}$.

Conditions: The integral is definite with bounds $u=0$ to $u=\ln 2$.; The integrand is $e^u \sqrt{1+e^u}$.; Substitution $x = 1 + e^u$ is used.

To evaluate the definite integral $\int_0^{\ln 2} e^u \sqrt{1+e^u} \, du$, use the substitution $x = 1 + e^u$, which yields $dx = e^u \, du$. Convert the limits of integration from $u=0$ and $u=\ln 2$ to $x=2$ and $x=3$, transforming the integral into $\int_2^3 \sqrt{x} \, dx$. Evaluating the antiderivative $\frac{2}{3}x^{3/2}$ at these new bounds gives the result $2\sqrt{3} - \frac{4\sqrt{2}}{3}$.

Conditions: The integral is definite with bounds $u=0$ to $u=\ln 2$.; The integrand is $e^u \sqrt{1+e^u}$.; Substitution $x = 1 + e^u$ is used.

To derive $\varphi(p^k)$, we count the integers from 1 to $p^k$ that are coprime to $p^k$. An integer is not coprime to $p^k$ if and only if it shares a prime factor with $p^k$.

Conditions: $p$ is a prime number.; $k$ is a positive integer (implied by context).; The counting range is from 1 to $p^k$ inclusive.

To derive $\varphi(p^k)$, we count the integers from 1 to $p^k$ that are coprime to $p^k$. An integer is not coprime to $p^k$ if and only if it shares a prime factor with $p^k$.

Conditions: $p$ is a prime number.; $k$ is a positive integer (implied by context).; The counting range is from 1 to $p^k$ inclusive.