What is the integral of ((2x+5))/((x+2)) ?

The integral of ((2x+5))/((x+2)) is 2(x+2)+ln|x+2|+C

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integral of ((2x+5))/((x+2))

integral

\frac{( 2 x + 5 )}{( x + 2 )}

2 (x + 2) + ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{2 x + 5}{x + 2} d x

Apply u-substitution

: \int \frac{2 u + 1}{u} d u

= \int \frac{2 u + 1}{u} d u

Expand

\frac{2 u + 1}{u} : 2 + \frac{1}{u}

= \int 2 + \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 2 d u + \int \frac{1}{u} d u

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= 2 u + ln ∣ u ∣

Substitute back

u = x + 2

= 2 (x + 2) + ln ∣ x + 2 ∣

Add a constant to the solution

= 2 (x + 2) + ln ∣ x + 2 ∣ + C

sin (2 x + 1)

x + 16

\frac{1}{( 35 - x )}

\frac{1}{( x ^{2} - 4 x )}

(2 x + 3)^{4}

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