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

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

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

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

ln (∣ x ∣ ∣ x + 1 ∣) + C

Solution steps

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = x (x + 1)

= ln ∣ x (x + 1) ∣

Simplify

ln ∣ x (x + 1) ∣ : ln (∣ x ∣ ∣ x + 1 ∣)

= ln (∣ x ∣ ∣ x + 1 ∣)

Add a constant to the solution

= ln (∣ x ∣ ∣ x + 1 ∣) + C

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