What is the integral of (x-4)/(x^2+x) ?

The integral of (x-4)/(x^2+x) is 5ln|x+1|-4ln|x|+C

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integral of (x-4)/(x^2+x)

\int \frac{x - 4}{x ^{2} + x} d x

5 ln ∣ x + 1 ∣ - 4 ln ∣ x ∣ + C

Solution steps

\int \frac{x - 4}{x ^{2} + x} d x

Expand

\frac{x - 4}{x ^{2} + x} : \frac{x}{x ^{2} + x} - \frac{4}{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{x}{x ^{2} + x} d x - \int \frac{4}{x ^{2} + x} d x

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

\int \frac{4}{x ^{2} + x} d x = 4 (ln ∣ x ∣ - ln ∣ x + 1 ∣)

= ln ∣ x + 1 ∣ - 4 (ln ∣ x ∣ - ln ∣ x + 1 ∣)

Simplify

ln ∣ x + 1 ∣ - 4 (ln ∣ x ∣ - ln ∣ x + 1 ∣) : 5 ln ∣ x + 1 ∣ - 4 ln ∣ x ∣

= 5 ln ∣ x + 1 ∣ - 4 ln ∣ x ∣

Add a constant to the solution

= 5 ln ∣ x + 1 ∣ - 4 ln ∣ x ∣ + C

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What is the integral of (x-4)/(x^2+x) ?
The integral of (x-4)/(x^2+x) is 5ln|x+1|-4ln|x|+C