# What is the integral of #int (2 -1/x)dx# from 1/2 to 3?

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To find the integral of ( \int \left(2 - \frac{1}{x}\right) , dx ) from ( x = \frac{1}{2} ) to ( x = 3 ), you would first integrate the expression ( 2 - \frac{1}{x} ) with respect to ( x ) and then evaluate the result at the upper and lower limits of integration and subtract the lower limit result from the upper limit result.

Integrating ( 2 - \frac{1}{x} ) with respect to ( x ) gives ( 2x - \ln|x| ).

Then, evaluate ( 2x - \ln|x| ) at ( x = 3 ) and ( x = \frac{1}{2} ) and subtract the results:

[ \left[ 2(3) - \ln|3| \right] - \left[ 2\left(\frac{1}{2}\right) - \ln\left|\frac{1}{2}\right| \right] ]

[ = \left[ 6 - \ln(3) \right] - \left[ 1 - \ln(2) \right] ]

[ = 6 - \ln(3) - 1 + \ln(2) ]

[ = 5 - \ln\left(\frac{3}{2}\right) ]

Therefore, the integral of ( \int \left(2 - \frac{1}{x}\right) , dx ) from ( x = \frac{1}{2} ) to ( x = 3 ) is ( 5 - \ln\left(\frac{3}{2}\right) ).

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