Integral de exponencial y límite

, por dani

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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} }}