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

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

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

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

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

Solution steps

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

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

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

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

\int 1 d x = x

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

Add a constant to the solution

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

\int \frac{1}{x - 1 \cdot ( x ^{2} )} d x

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

\int (4 + x) d x

e^{\frac{1}{θ}}

n = 0 \sum \infty \frac{3}{n n}

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