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Most Important Selected Qs for JEE AdvancedMathematicsDefinite Integration

Let f: R (0, ) be a real valued function satisfying ₀^x t f(x-t) d t=e^ 2 x -2 x-1 , then which of the following is(are) correct?

Options

  1. AThe value of (f⁻¹ )^ (4) equals 1 8
  2. BDerivative of f(x) with respect to e^x at x=0 is equal to 8
  3. CThe value of _ x 0 f(x)-4 x equals 4
  4. DThe value of f(0) is equal to 4

Correct answer

D. The value of f(0) is equal to 4

Step-by-step solution

₀^x t f(x-t) d t=e^ 2 x -1 Using King aligned & ₀^x(x-t) f(t) d t=e^ 2 x -1 & x ₀^x f(t) d t- ₀^x t f(t) d t=e^ 2 x -1 aligned Differentiate both sides x f(x)+ ₀^x f(t) d t-x f(x)=2 e^ 2 x -2 f(x)=4 e^ 2 x f(0)=4

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