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

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

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

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

2 ln ∣ x ∣ - x - \frac{x ^{2}}{2} + C

Solution steps

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

Expand

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int 1 d x = x

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

= 2 ln ∣ x ∣ - x - \frac{x ^{2}}{2}

Add a constant to the solution

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

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