What is the integral of-1/(2x^{1/2)sqrt(-x+1)} ?

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

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integral of-1/(2x^{1/2)sqrt(-x+1)}

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

- arcsin (x) + C

Solution steps

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

Take the constant out:

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

= - \frac{1}{2} \cdot \int \frac{1}{x ^{\frac{1}{2}} - x + 1} d x

Apply u-substitution

: \int \frac{2}{- u ^{2} + 1} d u

= - \frac{1}{2} \cdot \int \frac{2}{- u ^{2} + 1} d u

Take the constant out:

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

= - \frac{1}{2} \cdot 2 \cdot \int \frac{1}{- u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{- u ^{2} + 1} d u = arcsin (u)

= - \frac{1}{2} \cdot 2 arcsin (u)

Substitute back

u = x^{\frac{1}{2}}

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

Simplify

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

= - arcsin (x)

Add a constant to the solution

= - arcsin (x) + C

y = (x^{2} + 1)^{x}

\frac{d}{d x} (\frac{5}{x ^{7}})

- 3 e^{4 t}

\int_{1}^{3} \frac{e ^{x}}{e ^{x} - 1} d x

\int 3 - t e^{- t} d t

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