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

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

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

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

\frac{1}{2} (x - 4 x + ln ∣ x ∣ + 1) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Simplify

\frac{1}{2} ((1 - x)^{2} - (2 x - ln ∣ x ∣)) : \frac{1}{2} (x - 4 x + ln ∣ x ∣ + 1)

= \frac{1}{2} (x - 4 x + ln ∣ x ∣ + 1)

Add a constant to the solution

= \frac{1}{2} (x - 4 x + ln ∣ x ∣ + 1) + C

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