What is the integral of 1/((x+2)^2*(1/x)) ?

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

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integral of 1/((x+2)^2*(1/x))

integral

\frac{1}{( x + 2 ) ^{2} \cdot ( \frac{1}{x} )}

ln ∣ x + 2 ∣ + \frac{2}{x + 2} + C

Solution steps

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

Simplify

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

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

Apply u-substitution

: \int \frac{u - 2}{u ^{2}} d u

= \int \frac{u - 2}{u ^{2}} d u

Expand

\frac{u - 2}{u ^{2}} : \frac{1}{u} - \frac{2}{u ^{2}}

= \int \frac{1}{u} - \frac{2}{u ^{2}} d u

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}{u} d u - \int \frac{2}{u ^{2}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{2}{u ^{2}} d u = - \frac{2}{u}

= ln ∣ u ∣ - (- \frac{2}{u})

Substitute back

u = x + 2

= ln ∣ x + 2 ∣ - (- \frac{2}{x + 2})

Simplify

= ln ∣ x + 2 ∣ + \frac{2}{x + 2}

Add a constant to the solution

= ln ∣ x + 2 ∣ + \frac{2}{x + 2} + C

(2 x)^{3}

\frac{1}{( x ^{2} + 2 x ) ^{\frac{1}{2}}}

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

\frac{v}{2}

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

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