What is the integral of (3x+1)/(x^3+x) ?

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

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integral of (3x+1)/(x^3+x)

\int \frac{3 x + 1}{x ^{3} + x} d x

ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 + 3 arctan (x) + C

Solution steps

\int \frac{3 x + 1}{x ^{3} + x} d x

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Add a constant to the solution

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

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