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

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

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integral of x/(x+x^2)

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

ln ∣ 1 + x ∣ + C

Solution steps

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

Simplify

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

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

= ln ∣ 1 + x ∣

Add a constant to the solution

= ln ∣ 1 + x ∣ + C

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