How to prove that ∫ 0 ∞ cos ⁡ ( α t ) e − λ t d t = λ α 2 + λ 2 using real methods?

Question

How to prove that       ∫    0          ∞        cos  ⁡  (  α  t  )      e          −      λ      t              d    t  =      λ                  α        2            +              λ        2             using real methods?

Mya Hurst · Accepted Answer

Explicitly, with the choice  u  =  cos  ⁡  α  t  ,    d  u  =  −  α  sin  ⁡  α  t    d  t  ,    d  v  =      e          −      λ      t        ,    v  =  −      λ          −      1            e          −      λ      t        ,the first integration by parts gives  I  =  ∫      e          −      λ      t        cos  ⁡  α  t    d  t  =  −      1    λ        e          −      λ      t        cos  ⁡  α  t  −      α    λ    ∫      e          −      λ      t        sin  ⁡  α  t  .Then, the second integration by parts with the choice  u  =  sin  ⁡  α  t  ,    d  u  =  α  cos  ⁡  α  t    d  t  ,    d  v  =      e          −      λ      t        ,    v  =  −      λ          −      1            e          −      λ      t        ,gives  I  =  −      1    λ        e          −      λ      t        cos  ⁡  α  t  +      α          λ      2            e          −      λ      t        sin  ⁡  α  t  −            α      2              λ      2        ∫      e          −      λ      t        cos  ⁡  α  t    d  t  .But this integral on the RHS is simply I, so we conclude      (    1    +                  α        2                    λ        2              )    I  =            e              −        λ        t                    λ      2            (    α    sin    ⁡    α    t    −    λ    cos    ⁡    α    t    )    ,which gives the value of the indefinite integral as  I  =            e              −        λ        t                            λ        2            +              α        2              (  α  sin  ⁡  α  t  −  λ  cos  ⁡  α  t  )  +  C  ,for some constant of integration C. Taking the limit as   t  →  ∞, noting that sinαt and cosαt are bounded and finite, gives      ∫          t      =      0        ∞        e          −      λ      t        cos  ⁡  α  t    d  t  =  I  (  ∞  )  −  I  (  0  )  =  0  −      (    −          λ                        λ          2                +                  α          2                      )    =      λ                  λ        2            +              α        2              ,as claimed.

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