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

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

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

\int (1 - x) d x

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

Solution steps

\int 1 - x d x

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 - x

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

Add a constant to the solution

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

e^{- 4 t} \times sin (2 t)

\frac{10}{3} x^{3} - 5 x^{2} - 11 x + 6

\int \frac{x}{100 x ^{2} - 1} d x

\frac{d y}{d x}, x = y^{18}

g (x) = 1 - x

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