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

The integral of ((x+3))/((x-1)) is x-1+4ln|x-1|+C

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integral of ((x+3))/((x-1))

integral

\frac{( x + 3 )}{( x - 1 )}

x - 1 + 4 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{x + 3}{x - 1} d x

Expand

\frac{x + 3}{x - 1} : \frac{x}{x - 1} + \frac{3}{x - 1}

Apply the Sum Rule:

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

= \int \frac{x}{x - 1} d x + \int \frac{3}{x - 1} d x

\int \frac{x}{x - 1} d x = x - 1 + ln ∣ x - 1 ∣

\int \frac{3}{x - 1} d x = 3 ln ∣ x - 1 ∣

= x - 1 + ln ∣ x - 1 ∣ + 3 ln ∣ x - 1 ∣

Simplify

= x - 1 + 4 ln ∣ x - 1 ∣

Add a constant to the solution

= x - 1 + 4 ln ∣ x - 1 ∣ + C

What is the integral of ((x+3))/((x-1)) ?
The integral of ((x+3))/((x-1)) is x-1+4ln|x-1|+C