What is the integral of (6x+5)/(3-2x) ?

The integral of (6x+5)/(3-2x) is 9/2-3x-7ln|3-2x|+C

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integral of (6x+5)/(3-2x)

\int \frac{6 x + 5}{3 - 2 x} d x

\frac{9}{2} - 3 x - 7 ln ∣ 3 - 2 x ∣ + C

Solution steps

\int \frac{6 x + 5}{3 - 2 x} d x

Expand

\frac{6 x + 5}{3 - 2 x} : \frac{6 x}{3 - 2 x} + \frac{5}{3 - 2 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 \frac{6 x}{3 - 2 x} d x + \int \frac{5}{3 - 2 x} d x

\int \frac{6 x}{3 - 2 x} d x = \frac{9}{2} - 3 x - \frac{9}{2} ln ∣ 3 - 2 x ∣

\int \frac{5}{3 - 2 x} d x = - \frac{5}{2} ln ∣ 3 - 2 x ∣

= \frac{9}{2} - 3 x - \frac{9}{2} ln ∣ 3 - 2 x ∣ - \frac{5}{2} ln ∣ 3 - 2 x ∣

Add similar elements:

- \frac{9}{2} ln ∣ - 2 x + 3 ∣ - \frac{5}{2} ln ∣ - 2 x + 3 ∣ = - 7 ln ∣ - 2 x + 3 ∣

= \frac{9}{2} - 3 x - 7 ln ∣ 3 - 2 x ∣

Add a constant to the solution

= \frac{9}{2} - 3 x - 7 ln ∣ 3 - 2 x ∣ + C

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What is the integral of (6x+5)/(3-2x) ?
The integral of (6x+5)/(3-2x) is 9/2-3x-7ln|3-2x|+C