I would like to integrate the following: (1) I = ∫ 0 t d τ ∫ 0 t d τ ′ e − γ | τ − τ ′ |

Question

I would like to integrate the following:                    (1)                    I        =                  ∫          0          t                d        τ                  ∫          0          t                d                  τ          ′                          e                      −            γ                          |                        τ            −                          τ              ′                                      |

zalitiaf · Accepted Answer

It seems you made a mistake in the estimation if the inner integral. I am doing things much slower than you did:                              ∫          0          t                          e                      −            γ                          |                        τ            −                          τ              ′                                      |                                              d                  τ          ′                                    =                  ∫          0          τ                          e                      −            γ            (            τ            −                          τ              ′                        )                                  d                  τ          ′                +                  ∫          τ          t                          e                      −            γ            (                          τ              ′                        −            τ            )                                  d                  τ          ′                                                  =                  e                      −            γ            τ                                                (                                                              e                              γ                τ                                      −            1                    γ                                      )                          +                  e                      γ            τ                                                (                                                              e                              −                γ                τ                                      −                          e                              −                γ                t                                              γ                                      )                                                            =                  1          γ                                      (                          2        −                  e                      −            γ            τ                          −                  e                      −            γ            (            t            −            τ            )                                                )

sedeln5w · Accepted Answer

Let  H  (  x  )  =      {                            1          ,                          x          &amp;gt;          0                                      0          ,                          x          ≤          0                        be a Heaviside function. Then                    I                            =                  ∫          0          t                d        τ                  ∫          0          t                d                  τ          ′                          e                      −            γ                          |                        τ            −                          τ              ′                                      |                                      =                                          =                  ∫          0          t                          [                      ∫            0            t                                (            H            (            τ            −                          τ              ′                        )                          e                              −                γ                (                τ                −                                  τ                  ′                                )                                      +            H            (                          τ              ′                        −            τ            )                          e                              −                γ                (                                  τ                  ′                                −                τ                )                                      )                    d                      τ            ′                    ]                d        τ        =                                          =                  ∫          0          t                          [                      ∫            0            τ                                e                          −              γ              (              τ              −                              τ                ′                            )                                d                      τ            ′                    +                      ∫            τ            t                                e                          −              γ              (                              τ                ′                            −              τ              )                                d                      τ            ′                    ]                d        τ        =                                          =                  ∫          0          t                          [                      e                          −              γ              τ                                ⋅                                                                  1                γ                                            e                                  γ                                      τ                    ′                                                              |                        0            τ                    +                      e                          γ              τ                                ⋅                                                                  (                −                                  1                  γ                                )                                            e                                  −                  γ                                      τ                    ′                                                              |                        τ            t                    ]                d        τ        =                                          =                  ∫          0          t                          [                                    1              −                              e                                  −                  γ                  τ                                                      γ                    +                                    1              −                              e                                  −                  γ                  t                                                            e                                  γ                  τ                                                      γ                    ]                d        τ        =                                          =                  ∫          0          t                          [                      2            γ                    −                      1            γ                                e                          −              γ              τ                                −                                    e                              −                γ                t                                      γ                                e                          γ              τ                                ]                d        τ        =                                          =                                                        2              γ                        τ            |                    0          t                +                                                        1                              γ                2                                                    e                              −                γ                τ                                      |                    0          t                −                                                                        e                                  −                  γ                  t                                                            γ                2                                                    e                              γ                τ                                      |                    0          t                =                                          =                              2            t                    γ                +                                            e                              −                γ                t                                      −            1                                γ            2                          −                              1            −                          e                              −                γ                t                                                          γ            2                          =                  2          γ                t        −                  2                      γ            2                          (        1        −                  e                      −            γ            t                          )

I would like to integrate the following: <mtable displaystyle="true"> <mlabeledtr> <mt

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