Let f and g be functions with g(x)=_0^x t,f(t),dt satisfying g(x^3) = x^6+x^7.

Question

Let $f$ and $g$ be functions with $g(x)=\displaystyle\int_0^x t\,f(t)\,dt$ satisfying $g(x^3) = x^6+x^7$. Then $\displaystyle\sum_{r=1}^{15} f(r^3)$ is equal to:

Accepted Answer

Let $u=x^3$: $g(u) = u^2+u^{7/3}$. Then $g'(u) = 2u+\dfrac73u^{4/3} = u\,f(u)$, so

$$f(u) = 2+\frac73u^{1/3}$$

$$f(r^3) = 2+\frac73r$$

$$\sum_{r=1}^{15}f(r^3) = \sum_{r=1}^{15}\left(2+\frac73r\right) = 30 + \frac73\cdot\frac{15\cdot16}{2} = 30+280 = 310$$

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