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

The integral of (3x+4)/(x(x+2)^2) is -1/(x+2)+ln|x|-ln|x+2|+C

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

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

- \frac{1}{x + 2} + ln ∣ x ∣ - ln ∣ x + 2 ∣ + C

Solution steps

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

Expand

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

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

\int \frac{4}{x ( x + 2 ) ^{2}} d x = ln ∣ x ∣ - ln ∣ x + 2 ∣ + \frac{2}{x + 2}

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

Simplify

= - \frac{1}{x + 2} + ln ∣ x ∣ - ln ∣ x + 2 ∣

Add a constant to the solution

= - \frac{1}{x + 2} + ln ∣ x ∣ - ln ∣ x + 2 ∣ + C

What is the integral of (3x+4)/(x(x+2)^2) ?
The integral of (3x+4)/(x(x+2)^2) is -1/(x+2)+ln|x|-ln|x+2|+C