What is the integral of (2x)(1-x^{-3}) ?

The integral of (2x)(1-x^{-3}) is x^2+2/x+C

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integral of (2x)(1-x^{-3})

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

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

Solution steps

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

Take the constant out:

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

= 2 \cdot \int x (1 - x^{- 3}) d x

Expand

x (1 - x^{- 3}) : x - \frac{1}{x ^{2}}

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

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

\int x d x = \frac{x ^{2}}{2}

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

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

Simplify

2 (\frac{x ^{2}}{2} - (- \frac{1}{x})) : x^{2} + \frac{2}{x}

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

Add a constant to the solution

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

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