Answer
$$
\begin{aligned} F(T) &=\int_{0}^{T} f(x) d x \\
&=\int_{0}^{T} k b^{x} d x \\
&=\frac{k}{\ln b}\left[b^{T}-1\right] .
\end{aligned}
$$
Work Step by Step
The instantaneous death rate for members of a population at time $x$ is given by:
$$
f(x)=kb^{x}
$$
and the number of individuals who survive to age $T$ is given by:
$$
F(T) =\int_{0}^{T} f(x) d x
$$
To find a formula for $F(T)$ , use the Fundamental Theorem as follows:
$$
\begin{aligned} F(T) &=\int_{0}^{T} f(x) d x \\ &=\int_{0}^{T} k b^{x} d x \\ &=\int_{0}^{T} k e^{(\ln b) x} d x \\ &=k \int_{0}^{T} e^{(\ln b) x} d x \\ &=\frac{k}{\ln b}\left(e^{(\ln b) T}-1\right) \\ &=\frac{k}{\ln b}\left[b^{T}-1\right] .
\end{aligned}
$$
