Find ∫ a b 1 ( x − a ) ( b − x ) 2 d x

Question

Find      ∫    a          b            1                  (        x        −        a        )        (        b        −        x        )            2        d  x

Hugo Bruce · Accepted Answer

Note the identity                    x        −        a                    b        −        a              +                    b        −        x                    b        −        a              =  1which inspires us to do this change of variable:                    x        −        a                    b        −        a              =      sin          2        ⁡  t  ,     t  ∈  (  0  ,            π      2        )  .Now,                     b        −        x                    b        −        a              =      cos          2        ⁡  t  . Thus  d  x  =  2  (  b  −  a  )  cos  ⁡  t  sin  ⁡  t  d  t  .Hence      ∫          a              b                  1              (        x        −        a        )        (        b        −        x        )              d  x  =      ∫          0                      π        2                                2        (        b        −        a        )        cos        ⁡        t        sin        ⁡        t                    (        b        −        a        )        cos        ⁡        t        sin        ⁡        t              d  t  =  π  .

Find <msubsup> ∫ a <mrow class="MJX-TeXAtom-ORD"> b </mro

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