Heaviside and trig function integral ∫ sin ⁡ ( 3 t ) θ ( t ) d t

Question

Heaviside and trig function integral   ∫  sin  ⁡  (  3  t  )  θ  (  t  )  d  t

Braylon Perez · Accepted Answer

Integrate by parts,                                   ∫        sin        ⁡        3        t        θ        (        t        )        d        t        =        −                  1          3                ∫        d        (        cos        ⁡        3        t        )        θ        (        t        )        d        t                            =                            −                  1          3                θ        (        t        )        cos        ⁡        3        t        +                  1          3                ∫        cos        ⁡        3        t        δ        (        t        )        d        t                            =                            −                  1          3                θ        (        t        )        cos        ⁡        3        t        +                  1          3                θ        (        t        )        cos        ⁡        (        0        )                            =                              1          3                θ        (        t        )        (        1        −        cos        ⁡        3        t        )        =                  2          3                θ        (        t        )                  sin          2                ⁡                              3            t                    2                    where   δ  (  t  ) is the Dirac delta function, the derivative of the Heaviside function   θ  (  t  ).

Heaviside and trig function integral ∫

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