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

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

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

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

2 ln ∣ x ∣ - ln ∣ x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

= 2 ln ∣ x ∣ - ln ∣ x + 1 ∣

Add a constant to the solution

= 2 ln ∣ x ∣ - ln ∣ x + 1 ∣ + C

\frac{d y}{d x}, x y = x^{7} y + 72

x \to - \infty lim (ln (x - 2))

y = (tan (\frac{1}{x}))^{s e c (\frac{1}{x})}

x \to 1 lim (\frac{x ^{2} - x}{1 - x})

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