Using laplace theorem find the answer

Here's the general approach.

For the first integral, assume . I'm also interpreting it to say



(otherwise the integral simply diverges).

Writing , take the Laplace transform to get





which is an elementary integral that can be computed by decomposing into partial fractions, or by employing a proper trigonometric substitution. I leave that calculation to you; you should end up with



and taking the inverse Laplace transform yields the answer,



Hopefully that should give you a decent idea of how this method works.

For the remaining integrals, you will need to introduce a parameter before proceeding. Here's what you can try:





For the fifth integral, you can make the observation that the given integrand already resembles the Laplace transform of :



Finally, for the last integral, you can use