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Question

Integrate the rational function 1x2x(12x)

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Solution

1x2x(12x)=12+12(2xx(12x))

Let 2xx(12x)=Ax+B(12x)

(2x)=A(12x)+Bx ........(1)

Substituting x=0 and 12 in equation (1), we obtain
A=2 and B=3

2xx(12x)=2x+312x

Substituting in equation (1), we obtain

1x2x(12x)=12+12{2x+3(12x)}

1x2x(12x)dx={12+12(2x+312x)}dx

=x2+log|x|+32(2)log|12x|+C

=x2+log|x|34log|12x|+C

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