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

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

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

\int_{0}^{3} \frac{3}{x ( x - 3 )} d x

diverges

Solution steps

\int_{0}^{3} \frac{3}{x ( x - 3 )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int_{0}^{3} \frac{1}{x ( x - 3 )} d x

Take the partial fraction of

\frac{1}{x ( x - 3 )} : - \frac{1}{3 x} + \frac{1}{3 ( x - 3 )}

= 3 \cdot \int_{0}^{3} - \frac{1}{3 x} + \frac{1}{3 ( x - 3 )} d x

Apply the Sum Rule:

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

= 3 (- \int_{0}^{3} \frac{1}{3 x} d x + \int_{0}^{3} \frac{1}{3 ( x - 3 )} d x)

\int_{0}^{3} \frac{1}{3 x} 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 of 3/(x(x-3)) ?
The integral from 0 to 3 of 3/(x(x-3)) is diverges