What is the integral of (2x+2)/(x^2+2x+3) ?

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

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integral of (2x+2)/(x^2+2x+3)

\int \frac{2 x + 2}{x ^{2} + 2 x + 3} d x

ln x^{2} + 2 x + 3 + C

Solution steps

\int \frac{2 x + 2}{x ^{2} + 2 x + 3} d x

Expand

\frac{2 x + 2}{x ^{2} + 2 x + 3} : \frac{2 x}{x ^{2} + 2 x + 3} + \frac{2}{x ^{2} + 2 x + 3}

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}{x ^{2} + 2 x + 3} d x + \int \frac{2}{x ^{2} + 2 x + 3} d x

\int \frac{2 x}{x ^{2} + 2 x + 3} d x = ln x^{2} + 2 x + 3 - 2 arctan (\frac{1}{2} (x + 1))

\int \frac{2}{x ^{2} + 2 x + 3} d x = 2 arctan (\frac{1}{2} (x + 1))

= ln x^{2} + 2 x + 3 - 2 arctan (\frac{1}{2} (x + 1)) + 2 arctan (\frac{1}{2} (x + 1))

Simplify

= ln x^{2} + 2 x + 3

Add a constant to the solution

= ln x^{2} + 2 x + 3 + C

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