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\mbox{This antiderivative allows a nice, quick rewrite of the integrand,} \ \ \mbox{which will then allow a thankfully smooth integration.} \ \ \mbox{Simplification later may be only slightly less smooth.}
\mbox{We have:}
\int \ \ { t - 9 t^2 } / \sqrt{t} dt \quad = \ \int \ \ ( t / \sqrt{t} - { 9 t ^2 } / \sqrt{t} ) dt
\qquad \qquad qquad \qquad qquad \qquad = \ \int \ \ ( t^{1/2} - 9 t^{3/2} ) dt
\qquad \qquad qquad \qquad qquad \qquad = \ 2/3 t^{3/2} - 9 ( 2/5 t^{5/2} ) + C.
\mbox{To simplify this now, factor out the numerical fractions using} \ \ \mbox{the LCM of their denominators and the GCD of their} \ \ \mbox{numerators, and factor out the common variables using their} \ \ \mbox{lowest common power:
\int \ \ { t - 9 t^2 } / \sqrt{t} dt \quad = 2/3 t^{3/2} - 9 ( 2/5 t^{5/2} ) + C
\qquad \qquad qquad \qquad qquad \qquad = \ 2/15 t^{3/2} (5) - \ 2/15 t^{3/2} ( 9 \cdot 3 t) + C.
\qquad \qquad qquad \qquad qquad \qquad = \ 2/15 t^{3/2} ( 5 - 27 t ) + C.
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\mbox{You Got It Exactly Right !! Bravo !!!}
