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

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

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

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

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

Solution steps

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

\frac{x ^{3} + x + 1}{x ^{3} + x} = \frac{1}{x ^{3} + x} + 1

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

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

\int 1 d x = x

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

Add a constant to the solution

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

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