Integral de exponencial y límite

Halla $c(t)$ sabiendo que $c^\prime(t)=-0.1 \cdot e^{-0.3t}$ y que $\lim\limits_{t \rightarrow \infty} c(t)=0$

SOLUCIÓN

$c(t)=\int -0.1 \cdot e^{-0.3t} dt = -0.1 \cdot \frac{1}{-0.3}\int -0.3 \cdot e^{-0.3t} dt = \frac{0.1}{0.3} e^{-0.3t} +K$

$c(t)=\frac{1}{3} e^{-0.3t} +K = \frac{1}{3 \cdot e^{0.3t} }+K$

$\lim\limits_{t \rightarrow \infty} c(t)=\lim\limits_{t \rightarrow \infty}\frac{1}{3 \cdot e^{0.3t} }+K = \frac{1}{\infty}+K = K$

$\lim\limits_{t \rightarrow \infty} c(t)=0 \longtightarrow k=0$

Por tanto $\fbox{c(t)=\dfrac{1}{3 \cdot e^{0.3t} }}$