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Question

Integrate the rational functions.
2x3(x21)(2x+3)dx.

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Solution

2x3(x21)(2x+3)dx=2x3(x1)(x+1)(2x+3)dx
Let 2x3(x1)(x+1)(2x+3)=A(x1)+B(x+1)+C(2x+3)
2x3(x1)(x+1)(2x+3)=A(2x+3)(x+1)+B(x1)(2x+3)+C(x1)(x+1)(x1)(x+1)(2x+3)2x3=A(2x2+3x+2x+3)+B(2x22x+3x)+C(x21)2x3=x2(2A+2B+C)+x(5A+B)+(3A3BC)

On comparing the coefficients of x2,x and constant term on both sides, we get

2A+2B+C=0......(i)5A+B=2B=25A....(ii)and 3A3BC=3......(iii)On putting the value of B in Eqs.(i) and (iii),we get2A+2(25A)+C=02A+410A+C=08A+C=4......(iv)and 3A3(25A)C=33A6+15AC=318AC=3......(v)On adding Eqs.(iv) and (v),we get10A=1A=110On putting the value of A in Eq.(ii),we getB=25(110)B=2+12=52On putting the values of A and B in Eq.(i),we get2(110)+2(52)+C=015+5+C=0C=245A=110,B=52 and C=2452x3(x21)(2x+3)dx=(1)10(x1)dx+521x+1dx24512x+3dx=110log|x1|+52log|x+1|245log|2x+3|2+C=52log|x+1|110log|x1|125log|2x+3|+C


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