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

The integral of (3x+2)/((x-3)^2) is 3ln|x-3|-(11)/(x-3)+C

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integral of (3x+2)/((x-3)^2)

\int \frac{3 x + 2}{( x - 3 ) ^{2}} d x

3 ln ∣ x - 3 ∣ - \frac{11}{x - 3} + C

Solution steps

\int \frac{3 x + 2}{( x - 3 ) ^{2}} d x

Expand

\frac{3 x + 2}{( x - 3 ) ^{2}} : \frac{3 x}{( x - 3 ) ^{2}} + \frac{2}{( x - 3 ) ^{2}}

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{3 x}{( x - 3 ) ^{2}} d x + \int \frac{2}{( x - 3 ) ^{2}} d x

\int \frac{3 x}{( x - 3 ) ^{2}} d x = 3 (ln ∣ x - 3 ∣ - \frac{3}{x - 3})

\int \frac{2}{( x - 3 ) ^{2}} d x = - \frac{2}{x - 3}

= 3 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - \frac{2}{x - 3}

Simplify

3 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - \frac{2}{x - 3} : 3 ln ∣ x - 3 ∣ - \frac{11}{x - 3}

= 3 ln ∣ x - 3 ∣ - \frac{11}{x - 3}

Add a constant to the solution

= 3 ln ∣ x - 3 ∣ - \frac{11}{x - 3} + C

What is the integral of (3x+2)/((x-3)^2) ?
The integral of (3x+2)/((x-3)^2) is 3ln|x-3|-(11)/(x-3)+C