What is the integral of (x+5)/(x+1) ?

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

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integral of (x+5)/(x+1)

\int \frac{x + 5}{x + 1} d x

4 ln ∣ x + 1 ∣ + x + C

Solution steps

\int \frac{x + 5}{x + 1} d x

\frac{x + 5}{x + 1} = \frac{4}{x + 1} + 1

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

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

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

\int 1 d x = x

= 4 ln ∣ x + 1 ∣ + x

Add a constant to the solution

= 4 ln ∣ x + 1 ∣ + x + C

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