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

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

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integral of (x-x^{-2})/(x^2+2x^{-1)}

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

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

Solution steps

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

Apply u-substitution

= \int \frac{1}{2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = x^{2} + 2 x^{- 1}

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

2 x^{- 1} = \frac{2}{x}

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

Add a constant to the solution

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

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