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

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

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

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

ln ∣ x ∣ + ln ∣ 2 + x ∣ + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Expand

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

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

= ln ∣ x ∣ + ln ∣ 2 + x ∣

Add a constant to the solution

= ln ∣ x ∣ + ln ∣ 2 + x ∣ + C

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