What is the integral of ((4x+3)/(2x+5)) ?

The integral of ((4x+3)/(2x+5)) is 2x-7/2 ln|2x+5|+C

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integral of ((4x+3)/(2x+5))

\int (\frac{4 x + 3}{2 x + 5}) d x

2 x - \frac{7}{2} ln ∣ 2 x + 5 ∣ + C

Solution steps

\int \frac{4 x + 3}{2 x + 5} d x

Long division

\frac{4 x + 3}{2 x + 5} : 2 - \frac{7}{2 x + 5}

= \int 2 - \frac{7}{2 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

= \int 2 d x - \int \frac{7}{2 x + 5} d x

\int 2 d x = 2 x

\int \frac{7}{2 x + 5} d x = \frac{7}{2} ln ∣ 2 x + 5 ∣

= 2 x - \frac{7}{2} ln ∣ 2 x + 5 ∣

Add a constant to the solution

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

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What is the integral of ((4x+3)/(2x+5)) ?
The integral of ((4x+3)/(2x+5)) is 2x-7/2 ln|2x+5|+C