What is the integral of (2x+5)/(5x+9) ?

The integral of (2x+5)/(5x+9) is 2/5 x+7/25 ln|5x+9|+C

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integral of (2x+5)/(5x+9)

\int \frac{2 x + 5}{5 x + 9} d x

\frac{2}{5} x + \frac{7}{25} ln ∣ 5 x + 9 ∣ + C

Solution steps

\int \frac{2 x + 5}{5 x + 9} d x

Expand

\frac{2 x + 5}{5 x + 9} : \frac{2 x}{5 x + 9} + \frac{5}{5 x + 9}

Apply the Sum Rule:

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

= \int \frac{2 x}{5 x + 9} d x + \int \frac{5}{5 x + 9} d x

\int \frac{2 x}{5 x + 9} d x = 2 (\frac{1}{5} x - \frac{9}{25} ln ∣ 5 x + 9 ∣)

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

= 2 (\frac{1}{5} x - \frac{9}{25} ln ∣ 5 x + 9 ∣) + ln ∣ 5 x + 9 ∣

Simplify

2 (\frac{1}{5} x - \frac{9}{25} ln ∣ 5 x + 9 ∣) + ln ∣ 5 x + 9 ∣ : \frac{2}{5} x + \frac{7}{25} ln ∣ 5 x + 9 ∣

= \frac{2}{5} x + \frac{7}{25} ln ∣ 5 x + 9 ∣

Add a constant to the solution

= \frac{2}{5} x + \frac{7}{25} ln ∣ 5 x + 9 ∣ + C

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What is the integral of (2x+5)/(5x+9) ?
The integral of (2x+5)/(5x+9) is 2/5 x+7/25 ln|5x+9|+C