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Question

Evaluate 3x+1(x21)(x1)dx

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Solution

3x+1(x21)(x1)
=3x+1(x1)2(x+1)
Consider 3x+1(x1)2(x+1)=Ax1+B(x1)2+Cx+1
3x+1=A(x1)(x+1)+B(x+1)+C(x1)2
Put x=14=2BB=2
Put x=13+1=4CC=24=12
Put x=01=A+B+C1=A+212A=1+212=112=12
A=12,B=2,C=12
3x+1(x1)2(x+1)=121x1+21(x1)2121x+1
Integrating both sides, we get
3x+1(x1)2(x+1)dx=12dxx1+2dx(x1)212dxx+1
=12log|x1|2x112log|x+1|+c
=12logx1x+12x1+c

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