What is the integral of x/(x-21) ?

The integral of x/(x-21) is x-21+21ln|x-21|+C

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integral of x/(x-21)

\int \frac{x}{x - 21} d x

x - 21 + 21 ln ∣ x - 21 ∣ + C

Solution steps

\int \frac{x}{x - 21} d x

Apply u-substitution

: \int \frac{u + 21}{u} d u

= \int \frac{u + 21}{u} d u

Expand

\frac{u + 21}{u} : 1 + \frac{21}{u}

= \int 1 + \frac{21}{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 1 d u + \int \frac{21}{u} d u

\int 1 d u = u

\int \frac{21}{u} d u = 21 ln ∣ u ∣

= u + 21 ln ∣ u ∣

Substitute back

u = x - 21

= x - 21 + 21 ln ∣ x - 21 ∣

Add a constant to the solution

= x - 21 + 21 ln ∣ x - 21 ∣ + C

\frac{d}{d x} ((x + 2)^{2} (x - 1))

\int_{0}^{1} x^{2} + 1 d x

\frac{1}{s ( ( s - 1 ) ^{2} + 1 )}

\frac{d}{d y} (\frac{y ^{2}}{4})

\frac{d ^{2}}{d x ^{2}} (x e^{- 3 x})

What is the integral of x/(x-21) ?
The integral of x/(x-21) is x-21+21ln|x-21|+C