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Unformatted text preview: Computing Laplace Transforms Evaluating the Laplace Integral Since the Laplace transform is defined by an integral it is clear that its effective computation will require some proficiency in the basic integration skills of the standard calculus. We will illustrate with a number of examples. Example 1 Let f ( x ) = x, x [0 , ). Then ( L f ) s = Z e- sx x dx. Integrals of this type can be evaluated using integration by parts , for which the standard formula is Z g ( x ) h ( x ) dx = G ( x ) h ( x )- Z G ( x ) h ( x ) dx, where G ( x ) denotes an antiderivative of g ( x ). Taking g ( x ) = e- sx , h ( x ) = x and using the standard limits for the Laplace integral we have ( L x ) s = Z e- sx x dx =- e- sx s x + 1 s Z e- sx 1 dx. It is clear from this that the integral is defined for s > 0 and, for such values of s we have ( L x ) s = (0) + 1 s 2- lim x e- sx + e- s = 1 s 2 (0 + 1) = 1 s 2 ....
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- Spring '10