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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - 2 u d u

= \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

: \frac{2}{3} u^{\frac{3}{2}}

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

Substitute back

u = 1 - w

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

Simplify

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

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

Add a constant to the solution

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

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