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

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

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

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

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

Solution steps

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

Take the partial fraction of

\frac{x ^{2} + 2}{x ^{3} + x} : \frac{2}{x} - \frac{x}{x ^{2} + 1}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

Add a constant to the solution

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

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