What is the integral from 5 to 6 of x/(x-5) ?

The integral from 5 to 6 of x/(x-5) is diverges

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integral from 5 to 6 of x/(x-5)

\int_{5}^{6} \frac{x}{x - 5} d x

diverges

Solution steps

\int_{5}^{6} \frac{x}{x - 5} d x

Apply u-substitution

= \int_{0}^{1} \frac{u + 5}{u} d u

Expand

\frac{u + 5}{u} : 1 + \frac{5}{u}

= \int_{0}^{1} 1 + \frac{5}{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_{0}^{1} 1 d u + \int_{0}^{1} \frac{5}{u} d u

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

\int_{0}^{1} \frac{5}{u} d u =

diverges

If part of the expression diverges, then entire expression diverges

= diverges

What is the integral from 5 to 6 of x/(x-5) ?
The integral from 5 to 6 of x/(x-5) is diverges