Practice problems on Lapalace transformation

Practice Problems on Unit - I (Laplace Transforms)
Find Laplace transform of the following functions:

eat −1
1. 1. f (t) = a

2. f (t) = (sint − cost)2

3. f (t) = sint sin 2t sin 3t

4. g(t) = e6t (t + 2)2
sin 2t
5. h(t) = t
{
sin(t − π /3), if t > π /3,
6. f (t) =
0, if t < π /3.

e−at t n−1
7. f (t) = (n−1)!

8. f (t) = e−t sin2 t

9. f (t) = t sin at

10. f (t) = t 2 e2t
e−3/(s+1)
11. If L( f (t)) = 1s e−1/s , prove that L(e−t f (3t)) = s+1

Answers
1
1. s(s−a) , s>0

2. 1
s − s2 +2
2
2, s > 0


3. 1 1
2 [ s2 +4 − s2 +36
3 2
+ s2 +16 ]

2(2s2 −22s+61)
4. (s−6)3

π
5. 2 − tan−1 2s
e−π s/3
6. s2 +1
1
7. (s+a)n

2
8. (s+1)(s2 +2s+1)

2as
9. (s2 +a2 )2



1