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

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

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

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

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

Solution steps

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

Simplify

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

= \int x - 2 a + \frac{a ^{2}}{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 x d x - \int 2 a d x + \int \frac{a ^{2}}{x} d x

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

\int 2 a d x = 2 a x

\int \frac{a ^{2}}{x} d x = a^{2} ln ∣ x ∣

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

Add a constant to the solution

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

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