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Question

The value of the integral 1x2x(12x)dx is
(where C is an arbitrary constant)

A
x4ln|x|+34ln|12x|+C
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B
x4+ln|x|43ln|12x|+C
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C
x2+ln|x|34ln|12x|+C
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D
x2+ln|x|+43ln|12x|+C
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Solution

The correct option is C x2+ln|x|34ln|12x|+C
It can be seen that the given integrand is not a proper rational expression, therefore
1x2x(12x)=1x2x2x2=12[(x2x2)+(2x)x2x2]=12+12[2xx(12x)]

Let
2xx(12x)=Ax+B(12x)(2x)=A(12x)+Bx(1)
Equating the coefficients of x and constant term, we obtain
2A+B=1A=2B=3

Therefore,
2xx(12x)=2x+312x
Substituting in equation (1), we obtain
1x2x(12x)=12+12[2x+3(12x)]1x2x(12x)dx =[12+12(2x+3(12x))]dx =x2+ln|x|+32(2)ln|12x|+C =x2+ln|x|34ln|12x|+C
Where C is an arbitary constant.

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