What is the integral of (3x)/(3x+5) ?

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

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integral of (3x)/(3x+5)

\int \frac{3 x}{3 x + 5} d x

x - \frac{5}{3} ln ∣ 3 x + 5 ∣ + C

Solution steps

\int \frac{3 x}{3 x + 5} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{x}{3 x + 5} d x

Long division

\frac{x}{3 x + 5} : \frac{1}{3} - \frac{5}{3 ( 3 x + 5 )}

= 3 \cdot \int \frac{1}{3} - \frac{5}{3 ( 3 x + 5 )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (\int \frac{1}{3} d x - \int \frac{5}{3 ( 3 x + 5 )} d x)

\int \frac{1}{3} d x = \frac{1}{3} x

\int \frac{5}{3 ( 3 x + 5 )} d x = \frac{5}{9} ln ∣ 3 x + 5 ∣

= 3 (\frac{1}{3} x - \frac{5}{9} ln ∣ 3 x + 5 ∣)

Simplify

3 (\frac{1}{3} x - \frac{5}{9} ln ∣ 3 x + 5 ∣) : x - \frac{5}{3} ln ∣ 3 x + 5 ∣

= x - \frac{5}{3} ln ∣ 3 x + 5 ∣

Add a constant to the solution

= x - \frac{5}{3} ln ∣ 3 x + 5 ∣ + C

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