What is the integral from 0 to 3/2 of (x-3)/(2x-3) ?

The integral from 0 to 3/2 of (x-3)/(2x-3) is diverges

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integral from 0 to 3/2 of (x-3)/(2x-3)

\int_{0}^{\frac{3}{2}} \frac{x - 3}{2 x - 3} d x

diverges

Solution steps

\int_{0}^{\frac{3}{2}} \frac{x - 3}{2 x - 3} d x

Expand

\frac{x - 3}{2 x - 3} : \frac{x}{2 x - 3} - \frac{3}{2 x - 3}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{\frac{3}{2}} \frac{x}{2 x - 3} d x - \int_{0}^{\frac{3}{2}} \frac{3}{2 x - 3} d x

\int_{0}^{\frac{3}{2}} \frac{x}{2 x - 3} d x =

diverges

If part of the expression diverges, then entire expression diverges

= diverges

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What is the integral from 0 to 3/2 of (x-3)/(2x-3) ?
The integral from 0 to 3/2 of (x-3)/(2x-3) is diverges