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

The integral of ((2-sqrt(x))^2)/(sqrt(x)) is -2/3 (2-sqrt(x))^3+C

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

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

- \frac{2}{3} (2 - x)^{3} + C

Solution steps

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

Apply u-substitution

= \int - 2 u^{2} d u

Take the constant out:

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

= - 2 \cdot \int u^{2} d u

Apply the Power Rule

= - 2 \cdot \frac{u ^{3}}{3}

Substitute back

u = 2 - x

= - 2 \cdot \frac{( 2 - x ) ^{3}}{3}

Simplify

- 2 \cdot \frac{( 2 - x ) ^{3}}{3} : - \frac{2}{3} (2 - x)^{3}

= - \frac{2}{3} (2 - x)^{3}

Add a constant to the solution

= - \frac{2}{3} (2 - x)^{3} + C

What is the integral of ((2-sqrt(x))^2)/(sqrt(x)) ?
The integral of ((2-sqrt(x))^2)/(sqrt(x)) is -2/3 (2-sqrt(x))^3+C