Find this improper integral?

Question

Find the improper integral of te^-st dt from 0 to b where the limit of b approaches to positive infinity.

Answer 1

Use integration in parts

_∫0^b t e^-st dt = -(1/s) t e^-st (from o to b) + (1/s)∫₀^b e ^-st dt = -(b/s) e ^-sb - (1/s²) e ^-sb (from 0 to b) =

                          -(b/s)e ^{- sb} - (1/s²) e ^-sb + 1/s²

Now, if you take limes b →∞ you will get:  ∫ = 1/s²  (answer)