By using the de?nition of T(a) and integrating by parts, show that: T(a) = (a - 1)T(a - 1), T > 1. Solution If f(t) = t then F(s) = L(t) = Z 8 0 te -st dt = t × -1 s e -st t=8 t=0 | {z } = 0 for t = 0 and t = 8 - Z 8 0 1 × -1 s e -st dt integrating by parts = - 1 s 2 e -st t=8 t=0 integrating again, noting three minus signs = 1 s 2 substituting limits t = 8 and t = 0 Exercise Use integration by parts to show that L (t 2 ) = (2/s) × L(t). Generalise this to L (t n ).

1. By using the de?nition of T(a) and integrating by parts, show that: T(a) = (a - 1)T(a - 1), T > 1.
Solution
If f(t) = t then F(s) = L(t) = Z 8 0 te -st dt = t × -1 s e -st t=8 t=0 | {z } = 0 for t = 0
and t = 8 - Z 8 0 1 × -1 s e -st dt integrating by parts = - 1 s 2 e -st t=8 t=0 integrating again,
noting three minus signs = 1 s 2 substituting limits t = 8 and t = 0 Exercise Use integration by
parts to show that L (t 2 ) = (2/s) × L(t). Generalise this to L (t n )