# \ #
# \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. #
# \ #
# \mbox{You Got It Exactly Right !! Bravo !!!} #
