Question

Question #aa131

Answer 1

`-(1-x)^4/4+C`

Explanation:

Substitute `u=1-x` and `(du)/dx[1-x]=-1`

This simplifies to:
`-int[u^3]du`
Power Rule:
`=-(u^4/4)`
Undo subtitution:
`=-(1-x)^4/4+C`

Answer 2

`-1/4*(1-x)^4 + C`

Explanation:

Two ways to solve this :
1) You can develop `(1-x)^3` and integrate term by term but that isn't really clean...

2) Remember the formula : `(f(g))' = g'*f'(g)` ?
In your case :
`f : u -> u^4` and `g : x -> 1-x`
`g'(x) = -1`

then `f(g(x)) = (1-x)^4`
`(f(g(x)))' = - 4*(1-x)^3`
And because you don't want the -4 factor, you divide each side by this and you get :

`int (1-x)^3 dx = [-1/4*(1-x)^4]`