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

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

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

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

ln ∣ x + 2 ∣ - \frac{2}{x + 2} + C

Solution steps

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

Expand

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

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

\int \frac{x}{x ^{2} + 4 x + 4} d x = ln ∣ x + 2 ∣ + \frac{2}{x + 2}

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

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

Simplify

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

= ln ∣ x + 2 ∣ - \frac{2}{x + 2}

Add a constant to the solution

= ln ∣ x + 2 ∣ - \frac{2}{x + 2} + C

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