QJEEM25Ma5eb9: JEE Main 2025 Mathematics - Class 12 JEEM25Ma5eb9 Solution: Let be a real valued continuous function defined on the…

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QJEEM25Ma5eb9: JEE Main 2025 Mathematics - Class 12 JEEM25Ma5eb9 Solution: Let be a real valued continuous function defined on the…

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int\limits_0^x t f(t) d t$ . If $g\left(x^3\right)=x^6+x^7$ , then value of $\sum\limits_{r=1}^{15} f\left(r^3\right)$ is :

Multiple Choice Options

A 270
B 340
C 310
D 320

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Accepted Answer

$\begin{aligned} g (x) = x 2 + x^{\frac{7}{3}} \\ g^{'} (x) = 2 x + \frac{7}{3} x^{\frac{4}{3}} \\ f (x) = \frac{g^{'} (x)}{x} \\ f (x) = 2 + \frac{7}{3} x^{\frac{1}{3}} \\ f (r^{3}) = 2 + \frac{7 r}{3} \\ \sum_{r = 1}^{15} (1 + \frac{7}{3} r) = 310 \end{aligned}$

QJEEM25Ma5eb9: JEE Main 2025 Mathematics - Class 12 JEEM25Ma5eb9 Solution: Let be a real valued continuous function defined on the…

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