What is the integral of (2x^2+4x+22)/(x^2+2x+10) ?

The integral of (2x^2+4x+22)/(x^2+2x+10) is 2x+2/3 arctan(1/3 (x+1))+C

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integral of (2x^2+4x+22)/(x^2+2x+10)

\int \frac{2 x ^{2} + 4 x + 22}{x ^{2} + 2 x + 10} d x

2 x + \frac{2}{3} arctan (\frac{1}{3} (x + 1)) + C

Solution steps

\int \frac{2 x ^{2} + 4 x + 22}{x ^{2} + 2 x + 10} d x

Expand

\frac{2 x ^{2} + 4 x + 22}{x ^{2} + 2 x + 10} : \frac{2 x ^{2}}{x ^{2} + 2 x + 10} + \frac{4 x}{x ^{2} + 2 x + 10} + \frac{22}{x ^{2} + 2 x + 10}

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 ^{2}}{x ^{2} + 2 x + 10} d x + \int \frac{4 x}{x ^{2} + 2 x + 10} d x + \int \frac{22}{x ^{2} + 2 x + 10} d x

\int \frac{2 x ^{2}}{x ^{2} + 2 x + 10} d x = 2 (- ln x^{2} + 2 x + 10 - \frac{8}{3} arctan (\frac{1}{3} (x + 1)) + x)

\int \frac{4 x}{x ^{2} + 2 x + 10} d x = 4 (\frac{1}{2} ln x^{2} + 2 x + 10 - \frac{1}{3} arctan (\frac{1}{3} (x + 1)))

\int \frac{22}{x ^{2} + 2 x + 10} d x = \frac{22}{3} arctan (\frac{1}{3} (x + 1))

= 2 (- ln x^{2} + 2 x + 10 - \frac{8}{3} arctan (\frac{1}{3} (x + 1)) + x) + 4 (\frac{1}{2} ln x^{2} + 2 x + 10 - \frac{1}{3} arctan (\frac{1}{3} (x + 1))) + \frac{22}{3} arctan (\frac{1}{3} (x + 1))

Simplify

2 (- ln x^{2} + 2 x + 10 - \frac{8}{3} arctan (\frac{1}{3} (x + 1)) + x) + 4 (\frac{1}{2} ln x^{2} + 2 x + 10 - \frac{1}{3} arctan (\frac{1}{3} (x + 1))) + \frac{22}{3} arctan (\frac{1}{3} (x + 1)) : 2 x + \frac{2}{3} arctan (\frac{1}{3} (x + 1))

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

Add a constant to the solution

= 2 x + \frac{2}{3} arctan (\frac{1}{3} (x + 1)) + C

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What is the integral of (2x^2+4x+22)/(x^2+2x+10) ?
The integral of (2x^2+4x+22)/(x^2+2x+10) is 2x+2/3 arctan(1/3 (x+1))+C