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

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

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

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

\frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C

Solution steps

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

Take the partial fraction of

\frac{x ^{2} + 2}{x ( x + 2 ) ^{2}} : \frac{1}{2 x} + \frac{1}{2 ( x + 2 )} - \frac{3}{( x + 2 ) ^{2}}

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

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

\int \frac{1}{2 ( x + 2 )} d x = \frac{1}{2} ln ∣ x + 2 ∣

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

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

Simplify

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{x + 2}

Add a constant to the solution

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C

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