What is the integral of (arcsin(sqrt(x)))/(sqrt(x(1-x))) ?

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

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integral of (arcsin(sqrt(x)))/(sqrt(x(1-x)))

\int \frac{arcsin ( x )}{x ( 1 - x )} d x

arcsin^{2} (x) + C

Solution steps

\int \frac{arcsin ( x )}{x ( 1 - x )} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int v d v

= 2 \cdot \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= 2 \cdot \frac{v ^{2}}{2}

Substitute back

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

Simplify

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

= arcsin^{2} (x)

Add a constant to the solution

= arcsin^{2} (x) + C

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