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

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

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

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

diverges

Solution steps

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

Apply u-substitution

: \int_{- 2}^{0} \frac{u + 2}{u} d u

= \int_{- 2}^{0} \frac{u + 2}{u} d u

Expand

\frac{u + 2}{u} : 1 + \frac{2}{u}

= \int_{- 2}^{0} 1 + \frac{2}{u} d u

Apply the Sum Rule:

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

= \int_{- 2}^{0} 1 d u + \int_{- 2}^{0} \frac{2}{u} d u

\int_{- 2}^{0} 1 d u = 2

\int_{- 2}^{0} \frac{2}{u} d u =

diverges

If part of the expression diverges, then entire expression diverges

= diverges

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