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

The integral of sqrt(1-\sqrt{v)} is -2(-2/5 (1-sqrt(v))^{5/2}+2/3 (1-sqrt(v))^{3/2})+C

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integral of sqrt(1-\sqrt{v)}

\int 1 - v d v

- 2 (- \frac{2}{5} (1 - v)^{\frac{5}{2}} + \frac{2}{3} (1 - v)^{\frac{3}{2}}) + C

Solution steps

\int 1 - v d v

Apply u-substitution

: \int - 2 (- u + 1) u d u

= \int - 2 (- u + 1) 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 + 1) u d u

Expand

(- u + 1) u : - u^{\frac{3}{2}} + u

= - 2 \cdot \int - u^{\frac{3}{2}} + u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 2 (- \int u^{\frac{3}{2}} d u + \int u d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

= - 2 (- \frac{2}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = 1 - v

= - 2 (- \frac{2}{5} (1 - v)^{\frac{5}{2}} + \frac{2}{3} (1 - v)^{\frac{3}{2}})

Add a constant to the solution

= - 2 (- \frac{2}{5} (1 - v)^{\frac{5}{2}} + \frac{2}{3} (1 - v)^{\frac{3}{2}}) + C

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What is the integral of sqrt(1-\sqrt{v)} ?
The integral of sqrt(1-\sqrt{v)} is -2(-2/5 (1-sqrt(v))^{5/2}+2/3 (1-sqrt(v))^{3/2})+C