IMPROPER INTEGRALS
267
32.58 Calculate L{e'}.
The last limit is valid when s > 1. Thus, L{e'} = l/(s - 1) (denned for s > 1).
32.59
Calculate L {cos t}.
By integration by parts (see Problem 28.9), we obtain
Thus, L{cost} =s/(s
2 + 1).
32.60
If L{f} and L{f'} are defined, show that L{f'} = -/(O) + sL{f}.
For L{f'}, we use integration by parts with u = e sl, dv=f'{t)dt. Th
used the basic hypothesis that
[Here, we have
267
32.58 Calculate L{e'}.
The last limit is valid when s > 1. Thus, L{e'} = l/(s - 1) (denned for s > 1).
32.59
Calculate L {cos t}.
By integration by parts (see Problem 28.9), we obtain
Thus, L{cost} =s/(s
2 + 1).
32.60
If L{f} and L{f'} are defined, show that L{f'} = -/(O) + sL{f}.
For L{f'}, we use integration by parts with u = e sl, dv=f'{t)dt. Th
used the basic hypothesis that
[Here, we have