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

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

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

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

2 x arcsin (x) + 2 1 - x + C

Solution steps

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

Simplify

\frac{1}{x} arcsin (x) : \frac{arcsin ( x )}{x}

= \int \frac{arcsin ( x )}{x} d x

Apply Integration By Parts

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

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

\int \frac{1}{1 - x} d x = - 2 1 - x

= 2 x arcsin (x) - (- 2 1 - x)

Simplify

= 2 x arcsin (x) + 2 1 - x

Add a constant to the solution

= 2 x arcsin (x) + 2 1 - x + C

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