Solution
= Solution
The <expectation> is $\mathbb E[X]=k\int_0^\infty te^{-kt}dt$. An <integration by parts>, with the boundary term $te^{-kt}$ tending to zero, gives
$$
\boxed{\mathbb E[X]=\int_0^\infty e^{-kt}dt=\frac1k.}
$$
= Solution
The <expectation> is $\mathbb E[X]=k\int_0^\infty te^{-kt}dt$. An <integration by parts>, with the boundary term $te^{-kt}$ tending to zero, gives
$$
\boxed{\mathbb E[X]=\int_0^\infty e^{-kt}dt=\frac1k.}
$$