Simplify integral with random functions and their derivatives

Question

Let $F$ be a function and $f=F'$ the derivative of $F$. Then we obviously have for arbitrary $a,b$: $$ \int_a^b f(x) F(x)^9 dx =\frac{F(b)^{10}-F(a)^{10}}{10}. $$ Is there a way to obtain this result in Mathematica? I know that there is a way to do symbolic integration and taking derivatives but I'm now quite sure how I can define $f$ as the derivative of $F$. I am thus looking for how to define $f,F$ in Mathematica in such a way that if I call Integrate[f[x]*F[x]^9,{x,a,b}] it will return me the result (F[b]^10-F[a]^10)/10

Answer 1

Not sure what you are asking about. if you let $u = F(x)$ then $du = F'(x)dx = f(x)dx$ and the integral becomes $$ \int f(x) F(x)^9dx = \int u^9du = \frac{u^{10}}{10}+C=\frac{F(x)^{10}}{10}+C. $$

UPDATE

In Mathematica, use

Integrate[F'[x]*F[x]^9, x] or Integrate[F'[x]*F[x]^9, {x,a,b}]

In a similar query, here is the result from WolframAlpha