Verified answer

cotx = cosx/sinx

y = integral of (sinx)^4*(cotx)^3

= integral of (sinx)^4*(cosx)^3/(sinx)^3

= integral of sinx*(cosx)^3

let cosx = u

sinxdx = du

so, y = integral of u^3 du

= (u^4)/4

= (1/4)(cosx)^4 + constant

(sinx)^4*(cotx)^3

= sin^4(x) * [cos^3(x) / sin^3(x)]

= cos^3(x) sin(x)

Let u = cos^3x

du = -sinx dx

-1∫ cos^3(x) (-sin(x)) dx

= -1/4 cos^4x + c