What is the integral of sqrt(2-\sqrt{x)} ?

The integral of sqrt(2-\sqrt{x)} is -2(-2/5 (2-sqrt(x))^{5/2}+4/3 (2-sqrt(x))^{3/2})+C

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integral of sqrt(2-\sqrt{x)}

\int 2 - x d x

- 2 (- \frac{2}{5} (2 - x)^{\frac{5}{2}} + \frac{4}{3} (2 - x)^{\frac{3}{2}}) + C

Solution steps

\int 2 - x d x

Apply u-substitution

: \int 2 - u \cdot 2 u d u

= \int 2 - u \cdot 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 2 - u u d u

Apply u-substitution

: \int - (- v + 2) v d v

= 2 \cdot \int - (- v + 2) v d v

Take the constant out:

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

= 2 (- \int (- v + 2) v d v)

Expand

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

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

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 v^{\frac{3}{2}} d v + \int 2 v d v))

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

\int 2 v d v = \frac{4}{3} v^{\frac{3}{2}}

= 2 (- (- \frac{2}{5} v^{\frac{5}{2}} + \frac{4}{3} v^{\frac{3}{2}}))

Substitute back

= 2 (- (- \frac{2}{5} (2 - x)^{\frac{5}{2}} + \frac{4}{3} (2 - x)^{\frac{3}{2}}))

Simplify

= - 2 (- \frac{2}{5} (2 - x)^{\frac{5}{2}} + \frac{4}{3} (2 - x)^{\frac{3}{2}})

Add a constant to the solution

= - 2 (- \frac{2}{5} (2 - x)^{\frac{5}{2}} + \frac{4}{3} (2 - x)^{\frac{3}{2}}) + C

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What is the integral of sqrt(2-\sqrt{x)} ?
The integral of sqrt(2-\sqrt{x)} is -2(-2/5 (2-sqrt(x))^{5/2}+4/3 (2-sqrt(x))^{3/2})+C