I can find solution on fourth page here but it is not what I want. I have earlier learnt how to solve this kind of problems with serie like below. Could someone explain the integral way of doing things? Look at the border and the integrating terms, they are different -- you cannot integrate directly but use some tricks. What are they and the theory?
This is what I learnt earlier with series:
$2\int_0^x t y(t) \; dt = x^2 + y(x)=2 \lim_{n\rightarrow 0}\sum_{k=1}^n y(\epsilon_k) \; \Delta x_k$
where $x\in \mathbb R$, $y(x)$ is a continuous function in $\mathbb R$ to which holds $2\int_0^x ty(t) \; dt =x^2 +y(x)$ so
$2\int_0^x t y(t) \; dt =2\lim_{n\rightarrow 0}^n \sum_{k=1}^n y\left(\frac{kx}{n}\right)\left(\frac{x}{n}\right).$
but the solution shows an integral method. What is it (the integral method) on the page 4 called (sorry not in English, it uses simple integral apparently)?
More on page 644, X4.4, in the foreign course book I have been reading.