Question

For x ∈ R, x ≠ 0, if y(x) is a differentiable function such that x∫y(t)dt for t ∈ [1, x] = (x + 1)∫ty(t)dt for t ∈ [1, x], then y(x) equals (where C is a constant)

(A) Cx³(1/e^x)

(B) (C/x²)e^-1/x

(C) (C/x)e^-1/x

(D) (C/x³)e^-1/x

Answer 1

Answer is (D) (C/x³)e^-1/x

On applying Newton-Leibniz rule, we get

On differentiating with respect to x, we get