Integration of $c(y^2)(1-y)^4$

Question

Could anyone please help with integrating $f(y)=cy^2(1-y)^4$? Where $c$ is a constant.

Answer 1

Hint: Substitute $u = 1 - y$, giving

$$-c\int (1 - u)^2 u^4 du$$

Now expand $(1 - u)^2$ and integrate term-by-term.

Answer 2

Note that

$f(y) = cy^2(1-y)^4 = cy^2(1-4y+6y^2-4y^3+y^4) = c(y^2 - 4y^3 + 6y^4 - 4y^5 + y^6)$.

So $\int f(y) = c\int (y^2 + 4y^3 + 6y^4 - 4y^5 + y^6) dy$,

and then you can integrate each part separately.