What is the integral of (2x+3)/(x+7) ?

The integral of (2x+3)/(x+7) is 2(x+7)-11ln|x+7|+C

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integral of (2x+3)/(x+7)

\int \frac{2 x + 3}{x + 7} d x

2 (x + 7) - 11 ln ∣ x + 7 ∣ + C

Solution steps

\int \frac{2 x + 3}{x + 7} d x

Apply u-substitution

: \int \frac{2 u - 11}{u} d u

= \int \frac{2 u - 11}{u} d u

Expand

\frac{2 u - 11}{u} : 2 - \frac{11}{u}

= \int 2 - \frac{11}{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{11}{u} d u

\int 2 d u = 2 u

\int \frac{11}{u} d u = 11 ln ∣ u ∣

= 2 u - 11 ln ∣ u ∣

Substitute back

u = x + 7

= 2 (x + 7) - 11 ln ∣ x + 7 ∣

Add a constant to the solution

= 2 (x + 7) - 11 ln ∣ x + 7 ∣ + C

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What is the integral of (2x+3)/(x+7) ?
The integral of (2x+3)/(x+7) is 2(x+7)-11ln|x+7|+C