does any one know how to do integration of 32(y^4+10y^2+1)^7(y^3+5y)dy, U=y^4+10y^2+1?

Question

im supposed to use substitution to solve this problem! please help, I already try but seeems like im doing something wrong!

Answer 1

32(y^4+10y^2+1)^7(y^3+5y)dy---(1):
Yes, U=y^4+10y^2+1
du/dy=4(y³+5y)
dy=du/4(y³+5y)---(2)
Substitute (2) into (1):
integral 32u^7(y³+5y)[du/4(y³+5y)]
=integral 8u^7 du
=u^8+c
=(y^4+10y^2+1)^8+c

Answer 2

∫32(y^4 + 10y^2 + 1)^7(y^3 + 5y)dy
let u = y^4 + 10y^2 + 1
du = (4y^3+20y)dy = 4(y^3+5y)dy
∫32(y^4 + 10y^2 + 1)^7(y^3 + 5y)dy =
8∫u^7du =
(8/8)u^8 + C =
(y^4 + 10y^2 + 1)^8 + C

Answer 3

yes
u=y^4+10y^2+1
du=(4y^3+20y)dy
=4(y^3+5y)dy

integral (32 u^7. 4du)
=16u^8
=16(y^4+10y^2+1)^8+C