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

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

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integral of (x^2-1)/((x^3+x))

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

- ln ∣ x ∣ + ln x^{2} + 1 + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= - ln ∣ x ∣ + ln x^{2} + 1

Add a constant to the solution

= - ln ∣ x ∣ + ln x^{2} + 1 + C

\frac{d}{d x} (x^{4} + 8 x^{2} - x)

\int_{2}^{3} (3 x^{2} + 2 x + 3) d x

y^{^{''}} + 11 y^{^{'}} + 24 y = 0

\frac{d}{d x} (\frac{\frac{d}{d x} ( x ^{5} - 5 )}{x + 1})

\int e^{- \frac{1}{4} x^{2}} x d x

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