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

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

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

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

x arcsin (\frac{x}{1 + x}) + arctan (x) - x + C

Solution steps

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

Apply Integration By Parts

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

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

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

= x arcsin (\frac{x}{1 + x}) - (- arctan (x) + x)

Simplify

= x arcsin (\frac{x}{1 + x}) + arctan (x) - x

Add a constant to the solution

= x arcsin (\frac{x}{1 + x}) + arctan (x) - x + C

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