Let f be a real valued continuous function defined on the positive real axis such that . If , then value of is : [2025]

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Solution

Q.
Let f be a real valued continuous function defined on the positive real axis such that $g (x) = \int_{0}^{x} t f (t) d t$ . If $g (x^{3}) = x^{6} + x^{7}$ , then value of $\sum_{r = 1}^{15} f (r^{3})$ is : [2025]

1 270

2 340

3 320

4 310

Ans.
(4)

We have, $g (x) = \int_{0}^{x} t f (t) d t$

$g (x) = x^{2} + x^{7 / 3} \Rightarrow g' (x) = 2 x + \frac{7}{3} x^{4 / 3}$

As, $f (x) = \frac{g' (x)}{x} = 2 + \frac{7}{3} x^{1 / 3} \Rightarrow f (r^{3}) = 2 + \frac{7 r}{3}$

$∴ \sum_{r = 1}^{15} (2 + \frac{7}{3} r) = 30 + \frac{7}{3} [1 + 2 + . . . + 15] = 310$ .

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