JEE Mathematics 2025 Question with Solution

Given:

• $g(x) = \int_0^x t f(t) \, dt$
• $g(x^3) = x^6 + x^7$

Find:

• $\sum_{r=1}^{15} f(r^3)$

By the Fundamental Theorem of Calculus,

Therefore,

Differentiate $g(x^3) = x^6 + x^7$ with respect to $x$:

Using the chain rule,

Now substitute $g'(x^3) = x^3 f(x^3)$:

So,

Dividing by $3x^5$,

Now evaluate the required sum:

Thus,

Using

we get

Therefore, the value of $\sum_{r=1}^{15} f(r^3)$ is $310$. Hence, the correct option is D.

Given:

• $g(x) = \int_0^x t f(t) \, dt$
• $g(x^3) = x^6 + x^7$

Find:

• $\sum_{r=1}^{15} f(r^3)$

A common incorrect step is to write

because the integrand of $g$ is $t f(t)$, not just $f(t)$. Since

we must first differentiate $g(x)$ itself.

From the Fundamental Theorem of Calculus,

Replacing $x$ by $x^3$ gives

Now differentiate the identity

with respect to $x$:

Substitute $g'(x^3) = x^3 f(x^3)$:

which simplifies to

Divide both sides by $3x^5$:

Hence,

So,

The first solution shown in the source omits the factor coming from $g'(x) = x f(x)$ and writes an inconsistent expression. The corrected working above matches the second approach and the final answer $310$. Therefore, the correct option is D.