Question: integrate (arcsin x dx)
I = int arcsin x dx
Let u = arcsin(x)
Then x = sin(u)
And, dx = cos(u) du
This gives us,
I = int u.cos(u) du
Doing integration by parts,
I = u.sin(u) - int sin(u) du
I = u.sin(u) + cos(u)
I = arcsin(x).x + √(1 - sin^2(u))
I = x.arcsin(x) + √(1 - x^2)