Let f be a twice differentiable non-negative function such that (f(x))^2=25+\int_{0}^{x} ((f( t ))^2+ (f^{'}( t )) ^2) dt. Then the mean of f (\log _{ e }(1)) , f (\log _{ e }(2)) , . . . . . ., f (\log _{ e }(625)) is equal to \_\_\_\_。 [23 Jan 2026 Shift 1]

Differential Equations Part 3

Section: Mathematics

Question : 5 of 34

Marks: +1, -0

Let f be a twice differentiable non-negative function such that (f(x))2=25+

∫

((f(t))2+(f′(t))2)dt. Then the mean of f(loge(1)),f(loge(2)),......,f(loge(625)) is equal to ____。

[23 Jan 2026 Shift 1]

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