What is the integral of (sqrt(1-\sqrt{x)})/(sqrt(x)) ?

The integral of (sqrt(1-\sqrt{x)})/(sqrt(x)) is -4/3 (1-sqrt(x))^{3/2}+C

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integral of (sqrt(1-\sqrt{x)})/(sqrt(x))

\int \frac{1 - x}{x} d x

- \frac{4}{3} (1 - x)^{\frac{3}{2}} + C

Solution steps

\int \frac{1 - x}{x} d x

Apply u-substitution

= \int - 2 u d u

Take the constant out:

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

= - 2 \cdot \int u d u

Apply the Power Rule

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

Substitute back

u = 1 - x

= - 2 \cdot \frac{2}{3} (1 - x)^{\frac{3}{2}}

Simplify

- 2 \cdot \frac{2}{3} (1 - x)^{\frac{3}{2}} : - \frac{4}{3} (1 - x)^{\frac{3}{2}}

= - \frac{4}{3} (1 - x)^{\frac{3}{2}}

Add a constant to the solution

= - \frac{4}{3} (1 - x)^{\frac{3}{2}} + C

What is the integral of (sqrt(1-\sqrt{x)})/(sqrt(x)) ?
The integral of (sqrt(1-\sqrt{x)})/(sqrt(x)) is -4/3 (1-sqrt(x))^{3/2}+C