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

The integral of 2(sqrt(x)-1/(sqrt(x)))^2 is x^2-4x+2ln|x|+C

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integral of 2(sqrt(x)-1/(sqrt(x)))^2

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

x^{2} - 4 x + 2 ln ∣ x ∣ + C

Solution steps

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

Take the constant out:

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

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

Expand

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

= 2 \cdot \int x - 2 + \frac{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

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

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

\int 2 d x = 2 x

\int \frac{1}{x} d x = ln ∣ x ∣

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

Simplify

2 (\frac{x ^{2}}{2} - 2 x + ln ∣ x ∣) : x^{2} - 4 x + 2 ln ∣ x ∣

= x^{2} - 4 x + 2 ln ∣ x ∣

Add a constant to the solution

= x^{2} - 4 x + 2 ln ∣ x ∣ + C

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What is the integral of 2(sqrt(x)-1/(sqrt(x)))^2 ?
The integral of 2(sqrt(x)-1/(sqrt(x)))^2 is x^2-4x+2ln|x|+C