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

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

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

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

ln ∣ x + 2 ∣ + C

Solution steps

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

Cancel the common factor:

x

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

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = x + 2

= ln ∣ x + 2 ∣

Add a constant to the solution

= ln ∣ x + 2 ∣ + C

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