I have a continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ with continuous derivative and for which is true that:
$$\begin{array}{rcccl}\int_0^1{f(x)\cdot f'(x)\, dx}=0&\iff&\frac{1}{2}[(f(x))^2]_0^1=0&\iff& (f(1))^2-(f(0))^2=0\\ \int_0^1{(f(x))^2\cdot f'(x)\, dx}=18&\iff&\frac{1}{3}[(f(x))^3]_0^1=18&\iff& (f(1))^3-(f(0))^3=54 \end{array}$$ Now I want to find $$\int_0^1(f(x))^4\cdot f'(x)\, dx,$$ which is the same as $$\frac{1}{5}[(f(x))^5]_0^1=\frac{1}{5}\big((f(1))^5-(f(0))^5\big)=\frac{1}{5}\bigg((f(1))^2\cdot (f(1))^3-(f(0))^2\cdot (f(0))^3\bigg)$$ But I am shucked and I don't know how to continue. Any ideas?
You have transformed the question to the following system of equations
$$\begin{cases} A^2- B^2 = 0 & \\ A^3 - B^3 = 54 &\end{cases}$$
Two equations, two unknowns. It is not too difficult to solve $A, B$ explicitly. Then
$$\frac{1}{5} (A^5-B^5)$$
should also be found easily (since you know $A, B$).