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

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

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

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

\frac{1}{2} ln ∣ x ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 4 + arctan (\frac{x}{2})) + C

Solution steps

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

Take the partial fraction of

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

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

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

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

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

Add a constant to the solution

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

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