What is #int (2x^3-2x-3 ) / (-x^2+ 5 x )#?

1 Answer
Bdub
Mar 17, 2016

#int-2x-10-3/(5x)+237/(5(5-x)#
#=-x^2-10x-(3/5)lnx-(237/5)ln(5-x) +C#

Explanation:

#int(2x^3-2x-3)/(-x^2+5x)=int(-2x-10 +(48x-3)/(-x^2+5x ))#--->divide first

#(48x-3)/(-x^2+5x) = A/x +B/(5-x) #-->Partial Fractions

#48x-3 = A(5-x)+Bx#

#48x-3=5A-Ax+Bx->B-A=48,5A=-3#

#A=-3/5, B= 237/5#
#int-2x-10-3/(5x)+237/(5(5-x)#
#=-x^2-10x-(3/5)lnx-(237/5)ln(5-x) +C#

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