What is the integral of (2x+1)/(x+3) ?

The integral of (2x+1)/(x+3) is 2(x+3)-5ln|x+3|+C

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integral of (2x+1)/(x+3)

\int \frac{2 x + 1}{x + 3} d x

2 (x + 3) - 5 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{2 x + 1}{x + 3} d x

Apply u-substitution

: \int \frac{2 u - 5}{u} d u

= \int \frac{2 u - 5}{u} d u

Expand

\frac{2 u - 5}{u} : 2 - \frac{5}{u}

= \int 2 - \frac{5}{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{5}{u} d u

\int 2 d u = 2 u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= 2 u - 5 ln ∣ u ∣

Substitute back

u = x + 3

= 2 (x + 3) - 5 ln ∣ x + 3 ∣

Add a constant to the solution

= 2 (x + 3) - 5 ln ∣ x + 3 ∣ + C

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What is the integral of (2x+1)/(x+3) ?
The integral of (2x+1)/(x+3) is 2(x+3)-5ln|x+3|+C