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

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

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

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

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

Solution steps

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

Expand

\frac{x ^{2} - x + 2}{x ^{3} + x} : \frac{x ^{2}}{x ^{3} + x} - \frac{x}{x ^{3} + x} + \frac{2}{x ^{3} + x}

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

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

\int \frac{x}{x ^{3} + x} d x = arctan (x)

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

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

Add similar elements:

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

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

Add a constant to the solution

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

x \to 0 - lim (\frac{14 x ^{4} - 14 x}{x})

\frac{\partial}{\partial x} (x^{3} y + sin (x y))

\frac{d}{d x} (3 x^{2} - 4 x - 5)

\frac{( - s - 1 )}{s ^{2} + s + 1}

\frac{d}{d x} (\frac{10}{x})

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