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

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

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

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

- ln ∣ x + 2 ∣ + 2 ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

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

Simplify

ln ∣ x + 2 ∣ - 4 (\frac{1}{2} ln ∣ x + 2 ∣ - \frac{1}{2} ln ∣ x ∣) : - ln ∣ x + 2 ∣ + 2 ln ∣ x ∣

= - ln ∣ x + 2 ∣ + 2 ln ∣ x ∣

Add a constant to the solution

= - ln ∣ x + 2 ∣ + 2 ln ∣ x ∣ + C

ln (x), 7

x \to - 1 + lim (4 - x)

\frac{d}{d x} (2 (x^{3} + 4)^{2})

\int (4 x - 5) d x

x^{7} (x^{2} - \frac{3}{x ^{3}})

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