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

The integral of (x^3+3x^2+6)/(x^2+2) is 1/2 x^2+3x-ln|x^2+2|+1+C

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

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

\frac{1}{2} x^{2} + 3 x - ln x^{2} + 2 + 1 + C

Solution steps

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

Expand

\frac{x ^{3} + 3 x ^{2} + 6}{x ^{2} + 2} : \frac{x ^{3}}{x ^{2} + 2} + \frac{3 x ^{2}}{x ^{2} + 2} + \frac{6}{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{3 x ^{2}}{x ^{2} + 2} d x + \int \frac{6}{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{3 x ^{2}}{x ^{2} + 2} d x = - 32 arctan (\frac{x}{2}) + 3 x

\int \frac{6}{x ^{2} + 2} d x = 32 arctan (\frac{x}{2})

= \frac{1}{2} x^{2} + 1 - ln x^{2} + 2 - 32 arctan (\frac{x}{2}) + 3 x + 32 arctan (\frac{x}{2})

Simplify

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

Add a constant to the solution

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

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

\int 4 + 3 x d x

\frac{\partial}{\partial x} (x + sin (y + z))

5 + 2 x - 7 x^{3}

\frac{d}{d x} (\frac{1}{2 x ^{2}} + \frac{x ^{4}}{16})

What is the integral of (x^3+3x^2+6)/(x^2+2) ?
The integral of (x^3+3x^2+6)/(x^2+2) is 1/2 x^2+3x-ln|x^2+2|+1+C