A rate of decay of a substance is prepotional to the amount \( A(t) \) remaining at any gren tine (a) write the differential equation that matels the deay (b) If \( A(D)=A_{0} \), prove that if the time talen for The amount of the substance to become \( \frac{1}{2} A_{0} \) is \( T \), Then \( A(t)=A_{0} e\left(\frac{\text { thn } 2}{T}\right) \) (c) Hence, prove that the time taken for the amount remaining to be reduced to \( \frac{1}{2} A_{0} \) is \( 4.32 T \)

\( \int \frac { \operatorname { cen } x d x } { \cos x ( \cos x - 1 ) } \)

Ercontrar \( f(t) \) si \( F(s)=\frac{5}{s^{2}+4}+\frac{20 s}{s^{2}+9} \)

Given the system of differential equations: \[ \frac{d \mathbf{x}}{d t}=A \mathbf{x} \] where \( A=\left(\begin{array}{ll}4 & 0 \\ 0 & 3\end{array}\right) \), what is the general solution for \( \mathbf{x}(t) \) ?

Encuentre el centroide de la región acotada por las curvas: \( y=2 x-4, y=2 \sqrt{x}, x=1 \)

Analizar en cada caso si es posible aplicar el Teorema de Stokes para calcular la circulación de \( \vec{F} \) a lo largo de \( C:\left\{\begin{array}{l}x^{2}+y^{2}=1 \\ z=0\end{array}\right. \) Hallar el valor de esa circulación. (i) \( \vec{F}=\left\langle\frac{-x}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}, \frac{-y}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}, \frac{-z}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}\right\rangle \)