# Prove that these integrals are equal. How to complete the proof?

$$\int_0^xf(u)(x-u)^2du=2\int_0^x\left(\int_0^{u_2}\left(\int_0^{u_1}f(t)dt\right)du_1\right)du_2$$

Ok, I derived both parts 2 times wrt $x$ and got equal integrals. But I'm suspicious whether it is enough or not. Since two integrals can be differ by constant, so equality of their derivatives implies equality of integrals. But since we derived two times, we have that equality of derivatives implies that initial expressions can be differed by $ax+b$. So how to show strictly that these integrals actually allow to add or to subtract the expressions of form $ax+b$?

• $du$ or $du_2$? Commented Feb 24, 2015 at 11:53
• If you have $F''(x)=G''(x)$ everywhere, what you need to show that $F=G$ is just that, for example $F(0)=G(0)$ and $F'(0)=G'(0)$. Commented Feb 24, 2015 at 11:59
• thanks, there is $du_2$ instead of $du$ on the right hand Commented Feb 24, 2015 at 14:55

There aren't many details, so I don't think that you can assume that $f$ is at least continuous. If it's not then LHS of the equality may not be differentiable.