Inverse Laplace Transform of a Constant

We can find inverse Laplace Transform of a constant function very easily we need to have knowledge about Laplace transform. Laplace transform has a wide application in mathematics but it is mainly used in Integration. The main application of Laplace transform is to convert frequency Domain signal to time domain signal .here we will see an example how we can find the Inverse Laplace Transform of a constant function.
Example 1: Find the inverse transform of F(S) = (S+2)(S+4)/ S2+S?
Solution: For solving this type of problem we need to follow some steps given below.
Step 1: For converting this in to inverse Laplace transform we will see if we can factorize the denominator of the function.
So we can write S2+S as S(S+1) now we can rewrite the equation as (S+2)(S+4)/S(S+1).
Step 2: Now we will write the equation in partial fraction form as,
(S+2)(S+4)/S(S+1) = A/S + B/S+1,
Now our task is to find the value of constant ‘A’ and ‘B’ for that we need to apply cover up method with the help of this method we can find the value of constant very simply just need to have knowledge about solving fraction.
Firstly we will find the value of ‘A’,
(S+2)(S+4)/S(S+1) = A/S,
‘S’ gets canceled with ‘S’ and we remain with,
A= (S+2)(S+4)/(S+1),
Put S =0 we will get,
A= 8,
Similarly for ‘B’ we can rewrite the equation as,
(S+2)(S+4)/S(S+1) = B/S+1,
S+1 gets cancel with S+1 and we remain with
B= (S+2)(S+4)/S,
put S=-1,
B= -3,
Step 3: We can rewrite our equation as,
8/S – 1/S+1.
Now we can change them in to time domain by taking the inverse as
8-et,
As in the above question the inverse Laplace transform of 1/S is 1 so we can say that Laplace transform of a constant is 1/S. 

Math Topics