# Compute E(Y^2) where Y has density prop to  y^3 sin(y^4) cos(y^5) 1(0<y<1)

# importance sampling, method #1:

M = 10^6
uniflist = runif(M)
xlist = uniflist^(1/6)
numlist = sin(xlist^4) * cos(xlist^5)
denomlist = sin(xlist^4) * cos(xlist^5) / xlist^2
print( mean(numlist) / mean(denomlist) )

# importance sampling, method #2:

M = 10^6
uniflist = runif(M)
xlist = uniflist^(1/4)
numlist = sin(xlist^4) * cos(xlist^5) * xlist^2
denomlist = sin(xlist^4) * cos(xlist^5)
print( mean(numlist) / mean(denomlist) )