Antiderivative of the Arcsine function
Define . Then, . From this, it follows that .
In order to calculate the antiderivative for the arcsine function, change the expression in terms of the variable to its equivalent in terms of in the integrand as follows:
Now, define , so that and also let , and from this, . With this definitions at hand
apply the technique of integration by parts to obtain:
Since by definition, , and also, , it follows that , therefore,