What is the integral of (5x+6)/(x+1) ?

The integral of (5x+6)/(x+1) is 5(x+1)+ln|x+1|+C

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integral of (5x+6)/(x+1)

\int \frac{5 x + 6}{x + 1} d x

5 (x + 1) + ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{5 x + 6}{x + 1} d x

Apply u-substitution

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

Expand

\frac{5 u + 1}{u} : 5 + \frac{1}{u}

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

\int 5 d u = 5 u

\int \frac{1}{u} d u = ln ∣ u ∣

= 5 u + ln ∣ u ∣

Substitute back

u = x + 1

= 5 (x + 1) + ln ∣ x + 1 ∣

Add a constant to the solution

= 5 (x + 1) + ln ∣ x + 1 ∣ + C

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What is the integral of (5x+6)/(x+1) ?
The integral of (5x+6)/(x+1) is 5(x+1)+ln|x+1|+C