the integral from 0 to 1/4 of arcsin(x^(1/2))dx.

Answer 1

let z = x^.5
then dz = .5 x^-.5 dx
so dx = .5 x^.5 dz = .5 z dz
so
.5 z arcsin(z) dz

.5 z^2/2 sin^-1(z) -(1/4)sin^-1(z) +(z/4)sqrt(1-z^2)

or

.5 x sin^-1(x^.5) - (1/4)sin^-1(x^.5) + (1/4)x^.5 sqrt(1-x)

zero when x = 0
put in x = 1/4

Answer 2

dz = .5 x^-.5 dx
so dx = 2 x^.5 dz = 2 z dz
so
2 z arcsin(z) dz

etc