Compute ∫ 0 ∞ cos ⁡ ( x ) x 2 + a 2 d x , for a ∈ R using the Laplace Transform.

Name: Compute ∫ 0 ∞ cos ⁡ ( x ) x 2 + a 2 d x , for a ∈ R using the Laplace Transform.
Uploaded: 2022-02-23T17:22:57+00:00
Description: Compute ∫ 0 ∞ cos ⁡ ( x ) x 2 + a 2 d x , for a ∈ R using the Laplace Transform.

Question

Compute             ∫              0                    ∞                            cos        ⁡        (        x        )                              x          2                +                  a          2                            d      x      , for   a  ∈      R   using the Laplace Transform.

Jovanni Salinas · Accepted Answer

∫    0    ∞              cos      ⁡      x      t              1      +              t        2                    d    t  =      π    2        e          −              |            x              |            To transform your integral into this form use   x  ↦  t  ⋅  a. Now we can begin by taking the laplace transform of the integral      L    (  I  )  =      ∫          0              ∞            (          ∫              0                    ∞                            cos        ⁡        (        x        t        )                    1        +                  t          2                                d        t    )        e          −      s      x              d    xThe next step is to interchange the limits (fubinis theorem). Since       |    I      |  , converges in the Riemann sense, so does I.                              L                (        I        )        =                  ∫                      0                                ∞                                    (                      ∫                          0                                      ∞                                                          cos              ⁡              (              x              t              )                                      1              +                              t                2                                                          e                          −              s              x                                                      d                    x          )                          d                t                            =                  ∫                      0                                ∞                                    1                      1            +                          t              2                                                (                      ∫                          0                                      ∞                                cos          ⁡          (          x          t          )                      e                          −              s              x                                                      d                    x          )                          d                t                                          =                  ∫                      0                                ∞                                    1                      1            +                          t              2                                                s                                    s              2                        +                          t              2                                                d                t            Where we used the laplace tranform of   cos  ⁡  (  ω  x  ) in the last transition. The last expression can easilly be solved by partial fractions      1          1      +              t        2                  s                  s        2            +              t        2              =      s          1      −              s        2                  (          1                        s          2                +                  t          2                      −          1              1        +                  t          2                      )  To obtain the final answer you have to take the inverse-laplace transform of your integral.                              |                I                  |                ≤                  ∫                      0                                ∞                                    ∫                      0                                ∞                                                e                          −              s              x                                            1            +                          t              2                                                d                t                          d                x        =                  ∫                      0                                ∞                                                [                          π              2                                      e                              −                s                x                                      ]                    0          ∞                          d                t                          d                x        =                              [                          −                  π                      2            s                                    e                      −            s            x                                                              ]                                0          ∞                =                  π                      2            s

Compute int_0^(oo) (cos(x))/(x^2+a^2)dx, for a in RR using the Laplace Transform.

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