Calculus, EXT2 C1 2009 HSC 1d Evaluate `int_2^5 (x-6)/(x^2 + 3x-4)\ dx.` (4 marks) --- 8 WORK AREA LINES (style=lined) --- Show Answers Only `2 ln (3/4)` Show Worked Solution `text(Using partial fractions:)` `(x-6)/(x^2 + 3x-4)` `=(x-6)/((x+4)(x-1))` `= A/(x+4) + B/(x-1)` `:. x-6` `=A(x-1)+B(x+4)` `text(When)\ \ x=1,\ \ 5B=-5\ =>B=-1` `text(When)\ \ x=-4,\ \ -5A=-10\ =>A=2` `:. int_2^5 (x-6)/(x^2 + 3x-4)\ dx` `= int_2^5 2/(x+4)\ dx-int_2^5 (dx)/(x-1)` `= [2 ln (x+4)]_2^5 -[ln(x-1)]_2^5` `= 2(ln 9-ln 6)-(ln4-ln1)` `= 2ln(3/2)-ln4` `= 2ln(3/2)-2ln2` `= 2ln(3/4)`