Evaluate the integral, Mathematics

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Example:  If c ≠ 0 , evaluate the subsequent integral.

1840_Evaluate the integral.png

Solution

Remember that you require converting improper integrals to limits as given,

2314_Evaluate the integral1.png

Here, do the integral, so evaluate the limit.

1153_Evaluate the integral2.png

Here, at this point, we've found to be careful. The value of c will influence our answer. We've previously assumed that c was non-zero, this time we need to worry regarding to the sign of c. If c is positive the exponential will be present at infinity. Conversely, if c is negative the exponential will go to zero.

Therefore, the integral is only convergent that is the limit exists and is finite provided c<0.  Under this case we find,

2434_Evaluate the integral3.png

= -(1/c), given c< 0 .........(2)

Here we remember about how to do these, let's calculate some Laplace transforms. We'll start off along with probably the simplest Laplace transform to calculate.


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