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Question

A function f : R® R satisfies f(x+y) = f(x) + f(y) for all x,y Î R and is continuous throughout the domain. If I₁ + I₂ + I₃ + I₄+ I₅ = 450, where I_n = n.₀òⁿ f(x) dx. Find f(x).

Joshi sir comment

according to the given condition f(nx) = n f(x)

₀òⁿ f(x) dx

consider x = ny so this integration will become ₀ò¹ f(ny) ndy = n²₀ò¹ f(y) dy = n² ₀ò¹ f(x) dx

now by using these conditions I₁ = 1³ ₀ò¹ f(x) dx similarly others

put these values and get the answer

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