Use a trig substitution. Let x=1sin w. Then dx = cos w dw. This enables us to get that 1-x^2 out from under the radical operator:
If x =sin w, then x^2 = (sin w)^2 and 1-x^2 = (cos w)^2
Then sqrt(1-x^2) becomes sqrt( (cos w)^2 ) = cos w.
Also, x^3 becomes (sin w)^3.
So the original integrand becomes (sin w)^3*cos w*(cos w dw)
Simplifying this, we get (sin w)^3 * (cos w)^2 dw
We can eliminate that (cos w)^2. It equals 1 - (sin w)^2. So the integrand becomes
(sin w)^3 * [ 1 - (sin w)^2 ] dw
or: (sin w)^3dw - (sin w)^5dw
There are integration formulas for (sin w)^n that you could apply here.
Please let me know whether the information shared above is sufficient for you to finish this definite integral. If not I'd be glad to step back in.
Note that if we use the subst. x = sin w, then the angle w = arcsin x.
You are integrating from zero (0) to one (1). What are the "w" limits of integration?
If x=0, w = arcsin 0 = 0.
If x=1, w = arcsin 1 = π/2
so, you see, we can completely eliminate the variable x in favor of the new substitute variable w.