What is the integral of (2x+3)/((x+2)(x^2+5)) ?

The integral of (2x+3)/((x+2)(x^2+5)) is

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integral of (2x+3)/((x+2)(x^2+5))

\int \frac{2 x + 3}{( x + 2 ) ( x ^{2} + 5 )} d x

- \frac{1}{9} ln ∣ x + 2 ∣ + \frac{1}{18} ln x^{2} + 5 + 5 \frac{2}{9} arctan (\frac{x}{5}) + \frac{2}{3 5} arctan (\frac{x}{5}) + C

Solution steps

\int \frac{2 x + 3}{( x + 2 ) ( x ^{2} + 5 )} d x

Expand

\frac{2 x + 3}{( x + 2 ) ( x ^{2} + 5 )} : \frac{2 x}{( x + 2 ) ( x ^{2} + 5 )} + \frac{3}{( x + 2 ) ( x ^{2} + 5 )}

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 ) ( x ^{2} + 5 )} d x + \int \frac{3}{( x + 2 ) ( x ^{2} + 5 )} d x

\int \frac{2 x}{( x + 2 ) ( x ^{2} + 5 )} d x = 2 (- \frac{2}{9} ln ∣ x + 2 ∣ + \frac{1}{9} (ln x^{2} + 5 + 5 arctan (\frac{x}{5})))

\int \frac{3}{( x + 2 ) ( x ^{2} + 5 )} d x = 3 (\frac{1}{9} ln ∣ x + 2 ∣ + \frac{1}{9} (- \frac{1}{2} ln x^{2} + 5 + \frac{2}{5} arctan (\frac{x}{5})))

= 2 (- \frac{2}{9} ln ∣ x + 2 ∣ + \frac{1}{9} (ln x^{2} + 5 + 5 arctan (\frac{x}{5}))) + 3 (\frac{1}{9} ln ∣ x + 2 ∣ + \frac{1}{9} (- \frac{1}{2} ln x^{2} + 5 + \frac{2}{5} arctan (\frac{x}{5})))

Simplify

2 (- \frac{2}{9} ln ∣ x + 2 ∣ + \frac{1}{9} (ln x^{2} + 5 + 5 arctan (\frac{x}{5}))) + 3 (\frac{1}{9} ln ∣ x + 2 ∣ + \frac{1}{9} (- \frac{1}{2} ln x^{2} + 5 + \frac{2}{5} arctan (\frac{x}{5}))) : - \frac{1}{9} ln ∣ x + 2 ∣ + \frac{1}{18} ln x^{2} + 5 + 5 \frac{2}{9} arctan (\frac{x}{5}) + \frac{2}{3 5} arctan (\frac{x}{5})

= - \frac{1}{9} ln ∣ x + 2 ∣ + \frac{1}{18} ln x^{2} + 5 + 5 \frac{2}{9} arctan (\frac{x}{5}) + \frac{2}{3 5} arctan (\frac{x}{5})

Add a constant to the solution

= - \frac{1}{9} ln ∣ x + 2 ∣ + \frac{1}{18} ln x^{2} + 5 + 5 \frac{2}{9} arctan (\frac{x}{5}) + \frac{2}{3 5} arctan (\frac{x}{5}) + C

What is the integral of (2x+3)/((x+2)(x^2+5)) ?
The integral of (2x+3)/((x+2)(x^2+5)) is