I'm working my way through a textbook on probability in which the following integral appears:
$$F(y)=\int_1^\infty y^{n-1}\lambda^ne^{-\lambda y}\frac{1}{(n-1)!}dy-\int_1^\infty y^{n-2}\lambda^{n-1}e^{-\lambda y}\frac{1}{(n-2)!}dy~~~~ (1)$$
The author performs the integration in a single step:
$$F(y)=\left[-\frac{1}{\lambda}\frac{1}{(n-1)!}(y^{n-1}\lambda^ne^{-\lambda y})\right]_1^\infty~~~~ (2)$$
I just can't follow how the author jumps from (1) to (2). Is there some sort of integration technique I should know about?
$$ \begin{align} \frac{d}{dy} \left(-\frac{y^{n-1} \lambda^n e^{-\lambda y}}{\lambda\left(n-1\right)!} \right) &= - \left(-\lambda\right)\frac{y^{n-1} \lambda^n e^{-\lambda y}}{\lambda\left(n-1\right)!} -\left(n-1\right)\frac{y^{n-2} \lambda^n e^{-\lambda y}}{\lambda\left(n-1\right)!} \\ &= \frac{y^{n-1} \lambda^{n} e^{-\lambda y}}{\left(n-1\right)!} -\frac{y^{n-2} \lambda^{n-1} e^{-\lambda y}}{\left(n-2\right)!}, \end{align} $$ so $$ \int_a^b dy\left[\frac{y^{n-1} \lambda^{n} e^{-\lambda y}}{\left(n-1\right)!} -\frac{y^{n-2} \lambda^{n-1} e^{-\lambda y}}{\left(n-2\right)!}\right] = \left[-\frac{y^{n-1} \lambda^n e^{-\lambda y}}{\lambda\left(n-1\right)!} \right]_a^b. $$