I have this integral:
$$\int_a^ye^{-t}(t-2)dt$$ And I have to study the behavior of this integral at varying of $y_{0}$
----> $a$ means $y_{0}$
and $y$ without solving it with classic rules.
For example I can say:
if $y_{0}>2$ and $y\rightarrow\infty$ the integral diverges to $+\infty$.
This is pretty correct on the solutions but I'm very confused.
Wich is the relations used to write this statement? How is related the integrand function with his primitive to estimate if the integral diverges or not?
Thank you, I know that maybe is not so clear, but maybe someone expert will understand my question. Thank you :)
$$\begin{split} \int_{y_0}^ye^{-t}(t-2)\mathrm{d}t=&-\int_{y_0}^y(t-2)\mathrm{d}e^{-t} \\ =&-(t-2)e^{-t}\bigg|_{y_0}^y+\int_{y_0}^ye^{-t}\mathrm{d}(t-2) \\ =&-(t-2)e^{-t}\bigg|_{y_0}^y+\int_{y_0}^ye^{-t}\mathrm{d}t \\ =&-(t-2)e^{-t}\bigg|_{y_0}^y-e^{-t}\bigg|_{y_0}^y \\ =&-(t-1)e^{-t}\bigg|_{y_0}^y \\ =&-(y-1)e^{-y}+(y_0-1)e^{-y_0} \end{split}$$ So if $y_0>2, y\rightarrow+\infty$, the answer is $e^{-2}$. I'm not sure I have solved your problem. And I don't understand "classic rules".