Compute the following improper integrals, if possible. Label each integral as a Type I or Type...

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Compute the following improper integrals, if possible. Label each integral as a Type I or Type II improper integral. 1 (i) S da. Repeat this evaluation with the substitution u = -2. (x - 2)² re dr. It will be useful to show that the integrand is an odd function, then split the integral accordingly. 6. (iii) te-st dt, SER, S > 0. L'Hopital's Rule will be required. This integral is an example of a Laplace Transform, an integral transform that is used widely in science and engineering. It takes a function f(t), t € [0, ∞o), and transforms it via integration into a function F(s). Using the notation to indicate that a Laplace transform is being performed, its definition is F(s) = C{f(t)} f(t) est dt. One of the key motivations behind the transform is that operations on f in t-space become simpler to do in s-space. In particular, boundary value problems involving f(t) become easier to solve when the differential equation and f are trans- formed into s-space. In this problem f(t) = t. = Compute the following improper integrals, if possible. Label each integral as a Type I or Type II improper integral. 1 (i) S da. Repeat this evaluation with the substitution u = -2. (x - 2)² re dr. It will be useful to show that the integrand is an odd function, then split the integral accordingly. 6. (iii) te-st dt, SER, S > 0. L'Hopital's Rule will be required. This integral is an example of a Laplace Transform, an integral transform that is used widely in science and engineering. It takes a function f(t), t € [0, ∞o), and transforms it via integration into a function F(s). Using the notation to indicate that a Laplace transform is being performed, its definition is F(s) = C{f(t)} f(t) est dt. One of the key motivations behind the transform is that operations on f in t-space become simpler to do in s-space. In particular, boundary value problems involving f(t) become easier to solve when the differential equation and f are trans- formed into s-space. In this problem f(t) = t. =

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