Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int_0^x \mathrm{t} f(\mathrm{t}) \mathrm{dt}$. If $\mathrm{g}\left(x^3\right)=x^6+x^7$, then value of $\sum_{r=1}^{15} f\left(\mathrm{r}^3\right)$ is :

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Aicharya · Accepted Answer

$g(x)=x^2+x^{7 / 3}$
$\begin{aligned} & g^{\prime}(x)=2 x+\frac{7}{3} x^{4 / 3} \\ & f(x)=\frac{g^{\prime}(x)}{x} \\ & f(x)=2+\frac{7}{3} x^{1 / 3} \\ & f\left(r^3\right)=2+\frac{7}{3} r \\ & \sum_{r=1}^{15}\left(2+\frac{7}{3} r\right)=2(15)+\frac{7}{3}\left(\frac{15(16)}{2}\right)\end{aligned}$
$=310$

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int_0^x…

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