Let f   : R → R   be a function defined by f x =   x , x   ≤ 2 0 , x > 2 , where [ x ] is the greatest integer less than or equal to x . If I =   ∫ - 1 2 x f ( x 2 ) 2 + f ( x + 1 )   d x , then the value of ( 4 I - 1 ) is

Question

Let f &#160; : R &#8594; R &#160; be a function defined by f x = &#160; x , x &#160; &#8804; 2 0 , x &#62; 2 , where [ x ] is the greatest integer less than or equal to x . If I = &#160; &#8747; - 1 2 x f ( x 2 ) 2 + f ( x + 1 ) &#160; d x , then the value of ( 4 I - 1 ) is

Aicharya · Accepted Answer

f x + 1 = x + 1 , x + 1 &#8804; 2 0 , x + 1 &#62; 2 = x + 1 , x &#8804; 1 0 , x &#62; 1 = 0 , - 1 &#8804; x &#60; 0 1 , 0 &#8804; x &#60; 1 0 , 1 &#60; x &#60; 2 f x 2 = x 2 . x 2 &#8804; 2 0 , x 2 &#62; 2 = x 2 . | x | &#8804; 2 0 , | x | &#62; 2 = { 0, - 1 &#8804; x &#60; 0 0, 0 &#8804; x &#60; 1 1, 1 &#8804; x &#60; 2 0, 2 &#60; x &#60; 2 l = &#8747; - 1 0 0 d x + &#8747; 0 1 0 d x + &#8747; 1 2 x d x 2 + 0 + &#8747; 2 2 0 d x = 1 4 x 2 1 2 = 1 4 2 - 1 = 1 4 4 l = 1 4 l - 1 = 0

Let f   : R → R   be a function defined by f x =   x , x   ≤ 2 0 , x > 2…

Solution

Let f &#160; : R &#8594; R &#160; be a function defined by f x = &#160; x , x &#160; &#8804; 2 0 , x &#62; 2…

Solution

Let f : R → R be a function defined by f x = x , x ≤ 2 0 , x > 2…