What is the integral of (x^3+4x^2+2x)/(x^2+2) ?

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

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integral of (x^3+4x^2+2x)/(x^2+2)

\int \frac{x ^{3} + 4 x ^{2} + 2 x}{x ^{2} + 2} d x

\frac{1}{2} x^{2} + 4 x - 42 arctan (\frac{x}{2}) + 1 + C

Solution steps

\int \frac{x ^{3} + 4 x ^{2} + 2 x}{x ^{2} + 2} d x

Expand

\frac{x ^{3} + 4 x ^{2} + 2 x}{x ^{2} + 2} : \frac{x ^{3}}{x ^{2} + 2} + \frac{4 x ^{2}}{x ^{2} + 2} + \frac{2 x}{x ^{2} + 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} + 2} d x + \int \frac{4 x ^{2}}{x ^{2} + 2} d x + \int \frac{2 x}{x ^{2} + 2} d x

\int \frac{x ^{3}}{x ^{2} + 2} d x = \frac{1}{2} x^{2} + 1 - ln x^{2} + 2

\int \frac{4 x ^{2}}{x ^{2} + 2} d x = - 42 arctan (\frac{x}{2}) + 4 x

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

= \frac{1}{2} x^{2} + 1 - ln x^{2} + 2 - 42 arctan (\frac{x}{2}) + 4 x + ln x^{2} + 2

Simplify

= \frac{1}{2} x^{2} + 4 x - 42 arctan (\frac{x}{2}) + 1

Add a constant to the solution

= \frac{1}{2} x^{2} + 4 x - 42 arctan (\frac{x}{2}) + 1 + C

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