Question

Question #9387b

Answer 1

First simplify `100int_0^1x^9f(x^10)dx` through the following substitution: `u=x^10`, so `du=10x^9dx`.

`100int_0^1x^9f(x^10)dx=100/10int_0^1f(x^10)10x^9dx=10int_0^1f(u)du`

Note that the bounds did change, but not noticeably, since `0^10=0` and `1^10=1`.

Also note that `int_0^1f(u)du=int_0^1f(x)dx`.

So, we see the value of `"....."` is

`"....."+100int_0^1x^9f(x^10)dx="....."+10int_0^1f(x)dx="....."+50`

I'm not really sure what the `"....."` is supposed to stand for in the context of this problem.