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

The integral of (2-x)/(x^2+x) is 2ln|x|-3ln|x+1|+C

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

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

2 ln ∣ x ∣ - 3 ln ∣ x + 1 ∣ + C

Solution steps

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

Expand

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

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

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

\int \frac{x}{x ^{2} + x} d x = ln ∣ x + 1 ∣

= 2 (ln ∣ x ∣ - ln ∣ x + 1 ∣) - ln ∣ x + 1 ∣

Simplify

2 (ln ∣ x ∣ - ln ∣ x + 1 ∣) - ln ∣ x + 1 ∣ : 2 ln ∣ x ∣ - 3 ln ∣ x + 1 ∣

= 2 ln ∣ x ∣ - 3 ln ∣ x + 1 ∣

Add a constant to the solution

= 2 ln ∣ x ∣ - 3 ln ∣ x + 1 ∣ + C

What is the integral of (2-x)/(x^2+x) ?
The integral of (2-x)/(x^2+x) is 2ln|x|-3ln|x+1|+C