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99 Percentile Qs Bank for JEE MainMathematicsDefinite Integration

If f(x)= | x| x^2 , then _ 1 / e ^e f(x) d x=

Options

  1. Ae
  2. B1- 1 e
  3. Ce^2 (1- 1 e )
  4. D2 (1- 1 e )

Correct answer

D. 2 (1- 1 e )

Step-by-step solution

We have, f(x)= | x| x^2 = cases - x x^2 , & 0 < x < 1 x x^2 , & 1 < x < cases Now, as 1 e =0.367 and e 2.7182 aligned _ 1 / e ^e f(x) d x & = _ 1 / e ^1 f(x) d x+ ₁^e f(x) d x & = _ 1 / e ^1 - Iog x x^2 d x+ ₁^e Iog x x^2 d x aligned Let I= x x^2 d x= Iog _ I x x_ II ⁻² d x aligned & = x x⁻¹ (-1) - 1 x x⁻²⁺¹ -2+1 d x & =- x x + 1 x^2 d x= - x x - 1 x +C aligned Now, _ 1 / e ^e f(x) d x=- [ - Iog x x - 1 x ]_ 1 / e ^1+ [ - x x - 1 x ]₁^e aligned & = [(0+1)- (e e⁻¹+e ) ]- [ ( e e + 1 e )-(0+1) ] & =(1+e-e)- ( 1 e + 1

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