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

The integral of (x+5)/((x+3)(x-2)^2) is 2/25 ln|x+3|-2/25 ln|x-2|-2/(x-2)+3/(5(x-2))+C

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

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

\frac{2}{25} ln ∣ x + 3 ∣ - \frac{2}{25} ln ∣ x - 2 ∣ - \frac{2}{x - 2} + \frac{3}{5 ( x - 2 )} + C

Solution steps

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

Expand

\frac{x + 5}{( x + 3 ) ( x - 2 ) ^{2}} : \frac{x}{( x + 3 ) ( x - 2 ) ^{2}} + \frac{5}{( x + 3 ) ( x - 2 ) ^{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{x}{( x + 3 ) ( x - 2 ) ^{2}} d x + \int \frac{5}{( x + 3 ) ( x - 2 ) ^{2}} d x

\int \frac{x}{( x + 3 ) ( x - 2 ) ^{2}} d x = - \frac{1}{x - 2} - 3 (\frac{1}{25} ln ∣ x + 3 ∣ - \frac{1}{25} ln ∣ x - 2 ∣ - \frac{1}{5 ( x - 2 )})

\int \frac{5}{( x + 3 ) ( x - 2 ) ^{2}} d x = \frac{1}{5} ln ∣ x + 3 ∣ - \frac{1}{5} ln ∣ x - 2 ∣ - \frac{1}{x - 2}

= - \frac{1}{x - 2} - 3 (\frac{1}{25} ln ∣ x + 3 ∣ - \frac{1}{25} ln ∣ x - 2 ∣ - \frac{1}{5 ( x - 2 )}) + \frac{1}{5} ln ∣ x + 3 ∣ - \frac{1}{5} ln ∣ x - 2 ∣ - \frac{1}{x - 2}

Simplify

- \frac{1}{x - 2} - 3 (\frac{1}{25} ln ∣ x + 3 ∣ - \frac{1}{25} ln ∣ x - 2 ∣ - \frac{1}{5 ( x - 2 )}) + \frac{1}{5} ln ∣ x + 3 ∣ - \frac{1}{5} ln ∣ x - 2 ∣ - \frac{1}{x - 2} : \frac{2}{25} ln ∣ x + 3 ∣ - \frac{2}{25} ln ∣ x - 2 ∣ - \frac{2}{x - 2} + \frac{3}{5 ( x - 2 )}

= \frac{2}{25} ln ∣ x + 3 ∣ - \frac{2}{25} ln ∣ x - 2 ∣ - \frac{2}{x - 2} + \frac{3}{5 ( x - 2 )}

Add a constant to the solution

= \frac{2}{25} ln ∣ x + 3 ∣ - \frac{2}{25} ln ∣ x - 2 ∣ - \frac{2}{x - 2} + \frac{3}{5 ( x - 2 )} + C

What is the integral of (x+5)/((x+3)(x-2)^2) ?
The integral of (x+5)/((x+3)(x-2)^2) is 2/25 ln|x+3|-2/25 ln|x-2|-2/(x-2)+3/(5(x-2))+C