If \( \int_{0}^{x^{2}(1+x)} f(t) \mathrm{d} t=x \), then the value of \( f(2) \) is

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If \( \int_{0}^{x^{2}(1+x)} f(t) \mathrm{d} t=x \), then the value of \( f(2) \) is

MathematicsDefinite IntegrationJEE Main

Options:

A \( 1 / 2 \)
B \( 1 / 3 \)
C \( 1 / 4 \)
D \( 1 / 5 \)

Solution:

2040 Upvotes Verified Answer

The correct answer is: \( 1 / 5 \)

Differentiating both sides w.r.t. $x$ , we get

$f (x^{2} (1 + x)) \times (2 x + 3 x^{2}) = 1$ (Use Newton-Leibniz Rule for differentiation)

At $x = 1 \Rightarrow f (2) = \frac{1}{5}$

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