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

The integral of arccos(sqrt(x/(x+1))) is xarccos(sqrt(x/(x+1)))-arctan(sqrt(x))+sqrt(x)+C

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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