Why we can not prove the following by use of integration by parts in probability theory?

Question

We know that the expectation of a non-negative random variable is written as follows $$E[x]=\int_0^{\infty}[1-F_X(x)]dx=\int_0^{\infty}xf_X(x)dx$$ I want to ask why we cannot prove through integration by parts that $\int_0^{\infty}xf_X(x)dx$ results in $\int_0^{\infty}[1-F_X(x)]dx$? Any help in this regard will be much appreciated. Thanks in advance.