Use the substitution \(x = u ^ { 2 } + 1\) to show that
$$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \int _ { p } ^ { q } \frac { 6 \mathrm {~d} u } { u ( 3 + 2 u ) }$$
where \(p\) and \(q\) are positive constants to be found.
Hence, using algebraic integration, show that
$$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \ln a$$
where \(a\) is a rational constant to be found.
(6)
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