What is the integral of (x^3+3x^2+6x+5)/(x^2+x+2) ?

The integral of (x^3+3x^2+6x+5)/(x^2+x+2) is

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integral of (x^3+3x^2+6x+5)/(x^2+x+2)

\int \frac{x ^{3} + 3 x ^{2} + 6 x + 5}{x ^{2} + x + 2} d x

\frac{x ^{2}}{2} + 2 x + ln 4 x^{2} + 4 x + 8 + \frac{9}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{9}{7} arctan (\frac{1}{7} (2 x + 1)) + C

Solution steps

\int \frac{x ^{3} + 3 x ^{2} + 6 x + 5}{x ^{2} + x + 2} d x

Expand

\frac{x ^{3} + 3 x ^{2} + 6 x + 5}{x ^{2} + x + 2} : \frac{x ^{3}}{x ^{2} + x + 2} + \frac{3 x ^{2}}{x ^{2} + x + 2} + \frac{6 x}{x ^{2} + x + 2} + \frac{5}{x ^{2} + x + 2}

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{x ^{3}}{x ^{2} + x + 2} d x + \int \frac{3 x ^{2}}{x ^{2} + x + 2} d x + \int \frac{6 x}{x ^{2} + x + 2} d x + \int \frac{5}{x ^{2} + x + 2} d x

\int \frac{x ^{3}}{x ^{2} + x + 2} d x = \frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7})

\int \frac{3 x ^{2}}{x ^{2} + x + 2} d x = 3 (- \frac{1}{2} ln 4 x^{2} + 4 x + 8 - \frac{3}{7} arctan (\frac{1}{7} (2 x + 1)) + x)

\int \frac{6 x}{x ^{2} + x + 2} d x = 12 (\frac{1}{4} ln 4 x^{2} + 4 x + 8 - \frac{1}{2 7} arctan (\frac{2 x + 1}{7}))

\int \frac{5}{x ^{2} + x + 2} d x = \frac{10}{7} arctan (\frac{2 x + 1}{7})

= \frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7}) + 3 (- \frac{1}{2} ln 4 x^{2} + 4 x + 8 - \frac{3}{7} arctan (\frac{1}{7} (2 x + 1)) + x) + 12 (\frac{1}{4} ln 4 x^{2} + 4 x + 8 - \frac{1}{2 7} arctan (\frac{2 x + 1}{7})) + \frac{10}{7} arctan (\frac{2 x + 1}{7})

Simplify

\frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7}) + 3 (- \frac{1}{2} ln 4 x^{2} + 4 x + 8 - \frac{3}{7} arctan (\frac{1}{7} (2 x + 1)) + x) + 12 (\frac{1}{4} ln 4 x^{2} + 4 x + 8 - \frac{1}{2 7} arctan (\frac{2 x + 1}{7})) + \frac{10}{7} arctan (\frac{2 x + 1}{7}) : \frac{x ^{2}}{2} + 2 x + ln 4 x^{2} + 4 x + 8 + \frac{9}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{9}{7} arctan (\frac{1}{7} (2 x + 1))

= \frac{x ^{2}}{2} + 2 x + ln 4 x^{2} + 4 x + 8 + \frac{9}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{9}{7} arctan (\frac{1}{7} (2 x + 1))

Add a constant to the solution

= \frac{x ^{2}}{2} + 2 x + ln 4 x^{2} + 4 x + 8 + \frac{9}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{9}{7} arctan (\frac{1}{7} (2 x + 1)) + C

f^{^{'}} (x) = ln (x)

\int \frac{1}{x x ^{2} - 36} d x

\frac{\partial}{\partial y} (y - \frac{y}{x ^{2} + y ^{2}})

\frac{d N}{d x} = ((0.0922) N) (1 - \frac{N}{60})

\frac{d}{d x} ((e^{x} - e^{- 2 x})^{\frac{1}{5}})

What is the integral of (x^3+3x^2+6x+5)/(x^2+x+2) ?
The integral of (x^3+3x^2+6x+5)/(x^2+x+2) is