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KVPY2015MathematicsDefinite Integration

Define a function f: R R by f(x)= |x|,|x-1|, .|x-2 n| Where n is a fixed natural number, Then ₀^ 2 n f(x) d x is

Options

  1. An
  2. Bn ²
  3. C3 n
  4. D3 n ²

Correct answer

D. 3 n ²

Step-by-step solution

₀^ 2 n f(x) d x= ₀^ n |x-2 n| d x+ _ n ^ 2 n |x| d x ₀^ 2 n (2 n-x) d x+ _ n ^ 2 n x d x= (2 n x- x² 2 )₀^ n + ( x² 2 )_ n ^ 2 n =3 n²

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