How do I show the following: ∫ 0 α d x ∫ 0 α d y ln ⁡ | x − y | x y = 4 α ( ln ⁡ ( α ) + 2 ln ⁡ ( 2 ) − 3 )where α &gt; 0.

Question

How do I show the following:      ∫          0              α        d  x      ∫          0              α        d  y            ln      ⁡              |            x      −      y              |                    x      y        =  4  α  (  ln  ⁡  (  α  )  +  2  ln  ⁡  (  2  )  −  3  )where   α  &amp;gt;  0.

Harold Cantrell · Accepted Answer

Proceed as follows                              ∫                      0                                a                                    ∫                      0                                a                                                ln            ⁡                          |                        x            −            y                          |                                            x            y                          d        y                d        x                            =        2                  ∫                      0                                a                                    ∫                      0                                x                                                ln            ⁡            (            x            −            y            )                                x            y                          d        y                d        x                                                                              =                                      y              =              x                              t                2                                                    4                  ∫                      0                                a                                    ∫                      0                                1                          (        ln        ⁡        x        +        ln        ⁡        (        1        −                  t          2                )        )        d        t                d        x                                          =        4                  ∫                      0                                a                          (        ln        ⁡        x        +        2        ln        ⁡        2        −        2        )                d        x                                          =        4        a                (        ln        ⁡        a        +        2        ln        ⁡        2        −        3        )

How do I show the following: <msubsup> ∫ <mrow class="MJX-TeXAtom-ORD">

Answered question

Answer & Explanation

New Questions in Calculus 1