What is the integral of ((x^2+x))/((x-1)) ?

The integral of ((x^2+x))/((x-1)) is ((x-1)^2)/2+3(x-1)+2ln|x-1|+C

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integral of ((x^2+x))/((x-1))

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

\frac{( x - 1 ) ^{2}}{2} + 3 (x - 1) + 2 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{x ^{2} + x}{x - 1} d x

Apply u-substitution

: \int \frac{u ^{2} + 3 u + 2}{u} d u

= \int \frac{u ^{2} + 3 u + 2}{u} d u

Expand

\frac{u ^{2} + 3 u + 2}{u} : u + 3 + \frac{2}{u}

= \int u + 3 + \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 u d u + \int 3 d u + \int \frac{2}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int 3 d u = 3 u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{u ^{2}}{2} + 3 u + 2 ln ∣ u ∣

Substitute back

u = x - 1

= \frac{( x - 1 ) ^{2}}{2} + 3 (x - 1) + 2 ln ∣ x - 1 ∣

Add a constant to the solution

= \frac{( x - 1 ) ^{2}}{2} + 3 (x - 1) + 2 ln ∣ x - 1 ∣ + C

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What is the integral of ((x^2+x))/((x-1)) ?
The integral of ((x^2+x))/((x-1)) is ((x-1)^2)/2+3(x-1)+2ln|x-1|+C