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This is not a homework question, I am just trying to be come quicker at integrating by parts, when performing Laplace Transforms.

My textbook gives a basic example for performing the Laplace Transform of the variable t, to the transformed variable of s for the

equation: f(t)=t^2

It then provides this working for the solution:

Now, I do not understand how they have "evaluated the integral on the right hand side of the equation". The book provides no "list of integrals" and I have NO idea how they got this within a few lines? It seems as though there is some sort of almost quadratic they use to speed things up but I can't make out the rule.

I can do integration by parts, which takes a while, or I can use the method (example 9) a the bottom of this page

http://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts.aspx

which is much quicker. But if someone could please tell me how the textbook does it in so few lines that would be much appreciated.

Thanks