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99 Percentile Qs Bank for JEE MainMathematicsDefinite Integration

₀^ x x x+ x d x is equal to

Options

  1. A( -2) 2
  2. B+2 2
  3. C( +2) 2
  4. D-2 2

Correct answer

A. ( -2) 2

Step-by-step solution

₀^ x x x+ x d x Let I= ₀^ x x x+ x d x...(i) aligned _a^b f(x) d x & = _a^b f(a+b-x) d x I & = ₀^ ( -x) ( -x) ( -x)+ ( -x) d x I & = ₀^ ( -x)(- x) - x- x d x I & = ₀^ ( -x) x x+ x d x...(ii) aligned Adding Eqs. (i) and (ii), we get aligned 2 I & = ₀^ x x+( -x) x x+ x d x 2 I & = ₀^ x x+ x d x I & = 2 ₀^ x x+ x d x= 2 ₀^ x x 1 x + x x & = 2 ₀^ ( x 1+ x 1- x 1- x ) d x & = 2 ₀^ x(1- x) 1- ^2 x d x & = 2 ₀^ x- ^2 x ^2 x d x aligned aligned I= & 2 ₀^ ( x ^2 x - ^2 x ^2 x ) d x & 2 ₀^ ( x x- ^2 x ) d x & 2 ₀^ x x- ( ^2

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