What is the integral of (x^2+3x-5)/(x-1) ?

The integral of (x^2+3x-5)/(x-1) is ((x-1)^2)/2+5(x-1)-ln|x-1|+C

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integral of (x^2+3x-5)/(x-1)

\int \frac{x ^{2} + 3 x - 5}{x - 1} d x

\frac{( x - 1 ) ^{2}}{2} + 5 (x - 1) - ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{x ^{2} + 3 x - 5}{x - 1} d x

Apply u-substitution

: \int \frac{u ^{2} + 5 u - 1}{u} d u

= \int \frac{u ^{2} + 5 u - 1}{u} d u

Expand

\frac{u ^{2} + 5 u - 1}{u} : u + 5 - \frac{1}{u}

= \int u + 5 - \frac{1}{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 5 d u - \int \frac{1}{u} d u

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

\int 5 d u = 5 u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} + 5 u - ln ∣ u ∣

Substitute back

u = x - 1

= \frac{( x - 1 ) ^{2}}{2} + 5 (x - 1) - ln ∣ x - 1 ∣

Add a constant to the solution

= \frac{( x - 1 ) ^{2}}{2} + 5 (x - 1) - ln ∣ x - 1 ∣ + C

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What is the integral of (x^2+3x-5)/(x-1) ?
The integral of (x^2+3x-5)/(x-1) is ((x-1)^2)/2+5(x-1)-ln|x-1|+C