What is the integral of (x-4)/(x+1) ?

The integral of (x-4)/(x+1) is x+1-5ln|x+1|+C

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integral of (x-4)/(x+1)

\int \frac{x - 4}{x + 1} d x

x + 1 - 5 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{x - 4}{x + 1} d x

Expand

\frac{x - 4}{x + 1} : \frac{x}{x + 1} - \frac{4}{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{4}{x + 1} d x

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

\int \frac{4}{x + 1} d x = 4 ln ∣ x + 1 ∣

= x + 1 - ln ∣ x + 1 ∣ - 4 ln ∣ x + 1 ∣

Simplify

= x + 1 - 5 ln ∣ x + 1 ∣

Add a constant to the solution

= x + 1 - 5 ln ∣ x + 1 ∣ + C

What is the integral of (x-4)/(x+1) ?
The integral of (x-4)/(x+1) is x+1-5ln|x+1|+C