# Question #9387b

May 22, 2016

First simplify $100 {\int}_{0}^{1} {x}^{9} f \left({x}^{10}\right) \mathrm{dx}$ through the following substitution: $u = {x}^{10}$, so $\mathrm{du} = 10 {x}^{9} \mathrm{dx}$.

$100 {\int}_{0}^{1} {x}^{9} f \left({x}^{10}\right) \mathrm{dx} = \frac{100}{10} {\int}_{0}^{1} f \left({x}^{10}\right) 10 {x}^{9} \mathrm{dx} = 10 {\int}_{0}^{1} f \left(u\right) \mathrm{du}$

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

Also note that ${\int}_{0}^{1} f \left(u\right) \mathrm{du} = {\int}_{0}^{1} f \left(x\right) \mathrm{dx}$.

So, we see the value of $\text{.....}$ is

$\text{....."+100int_0^1x^9f(x^10)dx="....."+10int_0^1f(x)dx=".....} + 50$

I'm not really sure what the $\text{.....}$ is supposed to stand for in the context of this problem.