Integration of STP

\[\int STP = \delta \cdot e^2 \int \left(\left(\frac{x}{\tau}\right)^2 e^{-\frac{2}{\tau}x}\right) dx \]

IPP: we use \(u' = \frac{2x}{\tau^2}\) and \(v = -\frac{\tau}{2} \cdot e^{-\frac{2}{\tau}x}\)

\[\int STP = \delta \cdot e^2 \left( uv - \int u'v \right) + C\]

\[\int STP = \delta \cdot e^2 \left( \left(\frac{x}{\tau}\right)^2\cdot(-\frac{\tau}{2} \cdot e^{-\frac{2}{\tau}x}) - \int \frac{2x}{\tau^2} (-\frac{\tau}{2} \cdot e^{-\frac{2}{\tau}x})dx \right) + C\]

\[\int STP = \delta \cdot e^2 \left( -\left(\frac{x^2}{2\tau}\right) e^{-\frac{2}{\tau}x}+ \int \frac{x}{\tau} e^{-\frac{2}{\tau}x}dx \right) + C\]

Again, we use an IPP with $u’ = $ and \(v = -\frac{\tau}{2} \cdot e^{-\frac{2}{\tau}x}\)

\[\int STP = \delta \cdot e^2 \left( -\left(\frac{x^2}{2\tau}\right) e^{-\frac{2}{\tau}x}+ \left( uv - \int u'v \right) \right) + C\]

\[\int STP = \delta \cdot e^2 \left( -\left(\frac{x^2}{2\tau}\right) e^{-\frac{2}{\tau}x}+ \left( \frac{x}{\tau} (-\frac{\tau}{2} \cdot e^{-\frac{2}{\tau}x}) - \int \frac{1}{\tau} (-\frac{\tau}{2} \cdot e^{-\frac{2}{\tau}x}) \right) \right) + C\]

\[\int STP = \delta \cdot e^2 \left( -\left(\frac{x^2}{2\tau}\right) e^{-\frac{2}{\tau}x} -\frac{x}{2} \cdot e^{-\frac{2}{\tau}x} + \int \frac{1}{2} \cdot e^{-\frac{2}{\tau}x} \right) + C\]

\[\int STP = \delta \cdot e^2 \left( -\left(\frac{x^2}{2\tau}\right) e^{-\frac{2}{\tau}x} -\frac{x}{2} \cdot e^{-\frac{2}{\tau}x} -\frac{\tau}{4} \cdot e^{-\frac{2}{\tau}x} \right) + C\]

\[\int STP = \delta \cdot e^{(1-\frac{x}{\tau})2} \left( -\frac{x^2}{2\tau} -\frac{x}{2} -\frac{\tau}{4}\right) + C\]

\[\int_0^t STP = \frac{\delta \cdot e^{2} \cdot\tau}{4} - \delta \cdot e^{(1-\frac{t}{\tau})2} \left(\frac{t^2}{2\tau}+\frac{t}{2}+\frac{\tau}{4}\right) \]

\[\int_0^t LTP = t - \frac{1}{\lambda} (1-e^{-\lambda\cdot t})\]

\[\int_0^t prod = \theta_0\cdot t + (\theta_{\infty} - \theta_0)\left(t - \frac{1}{\lambda} (1-e^{-\lambda\cdot t})\right) + \theta_{\infty}\left(\frac{\delta \cdot e^{2} \cdot\tau}{4} - \delta \cdot e^{(1-\frac{t}{\tau})2} \left(\frac{t^2}{2\tau}+\frac{t}{2}+\frac{\tau}{4}\right) \right) \]

if \(mort = \theta_{\infty}\):

\[\int_0^t mort = \theta_{\infty}\cdot t\] then:

\[BA(t) = BA(0) + \theta_0\cdot t + (\theta_{\infty} - \theta_0)\left(t - \frac{1}{\lambda} (1-e^{-\lambda\cdot t})\right) + \theta_{\infty}\left(\frac{\delta \cdot e^{2} \cdot\tau}{4} - \delta \cdot e^{(1-\frac{t}{\tau})2} \left(\frac{t^2}{2\tau}+\frac{t}{2}+\frac{\tau}{4}\right) \right) - \theta_{\infty}\cdot t\]

\[BA(t) = BA(0) - \frac{\theta_{\infty} - \theta_0}{\lambda}(1-e^{-\lambda\cdot t}) + \theta_{\infty}\left(\frac{\delta \cdot e^{2} \cdot\tau}{4} - \delta \cdot e^{(1-\frac{t}{\tau})2} \left(\frac{t^2}{2\tau}+\frac{t}{2}+\frac{\tau}{4}\right)\right) \]