Question

How do you integrate `int x(x^2+1)^3dx` from [-1,1]?

Answer 1

We know that,
If function `f` is odd and continous in `[-a,a]`, then

`color(red)(int_-a^af(x)dx=0,ainR^+`

Explanation:

Here,

`I=int_-1^1 x(x^2+1)^3dx`

Please see the graph of `y=x(x^2+1)^3`
graph{x(x^2+1)^3 [-12.3, 12.3, -6.135, 6.175]}

Let, `f(x)=x(x^2+1)^3`

`f(-x)=(-x)((-x)^2+1)^3=-x(x^2+1)^3=-f(x)`

So,

`f(-x)=-f(x)=>f` is odd function,

and from the above graph we can say that

`f` is continous on [-1,1].

Hence,

`int_-1^1x(x^2+1)^3dx=0`