What is the integral of ((3x+4))/(x+3) ?

The integral of ((3x+4))/(x+3) is 3(x+3)-5ln|x+3|+C

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integral of ((3x+4))/(x+3)

\int \frac{( 3 x + 4 )}{x + 3} d x

3 (x + 3) - 5 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{3 x + 4}{x + 3} d x

Apply u-substitution

= \int \frac{3 u - 5}{u} d u

Expand

\frac{3 u - 5}{u} : 3 - \frac{5}{u}

= \int 3 - \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 3 d u - \int \frac{5}{u} d u

\int 3 d u = 3 u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= 3 u - 5 ln ∣ u ∣

Substitute back

u = x + 3

= 3 (x + 3) - 5 ln ∣ x + 3 ∣

Add a constant to the solution

= 3 (x + 3) - 5 ln ∣ x + 3 ∣ + C

What is the integral of ((3x+4))/(x+3) ?
The integral of ((3x+4))/(x+3) is 3(x+3)-5ln|x+3|+C