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

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

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

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

\frac{1}{2} ln \frac{x ^{2}}{2} + 1 + 2 arctan (\frac{x}{2}) + C

Solution steps

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

Apply Integral Substitution

= \int \frac{u + 2}{u ^{2} + 1} d u

Expand

\frac{u + 2}{u ^{2} + 1} : \frac{u}{u ^{2} + 1} + \frac{2}{u ^{2} + 1}

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{u}{u ^{2} + 1} d u + \int \frac{2}{u ^{2} + 1} d u

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

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

= \frac{1}{2} ln u^{2} + 1 + 2 arctan (u)

Substitute back

u = \frac{x}{2}

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

(\frac{x}{2})^{2} = \frac{x ^{2}}{2}

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

Add a constant to the solution

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

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