What is the integral from 1 to x of 2-(2x-2ln(x))/x ?

The integral from 1 to x of 2-(2x-2ln(x))/x is ln^2(x)

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integral from 1 to x of 2-(2x-2ln(x))/x

\int_{1}^{x} 2 - \frac{2 x - 2 ln ( x )}{x} d x

ln^{2} (x)

Solution steps

\int_{1}^{x} 2 - \frac{2 x - 2 ln ( x )}{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_{1}^{x} 2 d x - \int_{1}^{x} \frac{2 x - 2 ln ( x )}{x} d x

\int_{1}^{x} 2 d x = 2 x - 2

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

= 2 x - 2 - (2 x - 2 - ln^{2} (x))

Simplify

2 x - 2 - (2 x - 2 - ln^{2} (x)) : ln^{2} (x)

= ln^{2} (x)

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What is the integral from 1 to x of 2-(2x-2ln(x))/x ?
The integral from 1 to x of 2-(2x-2ln(x))/x is ln^2(x)