Let F x =   ∫ x x 2 + π 6 2 cos 2 ⁡ t   d t for all x   ∈ R   a n d   f   :   0 , 1 2 → 0 ,   ∞ be a continuous function. For a ∈ 0 , 1 2 , if F ′ a + 2 is the area of the region bounded by x = 0 ,   y = 0 …

Question

Let F x = &#160; &#8747; x x 2 + &#960; 6 2 cos 2 &#8289; t &#160; d t for all x &#160; &#8712; R &#160; a n d &#160; f &#160; : &#160; 0 , 1 2 &#8594; 0 , &#160; &#8734; be a continuous function. For a &#8712; 0 , 1 2 , if F &#8242; a + 2 is the area of the region bounded by x = 0 , &#160; y = 0 , &#160; y = f x &#160; a n d &#160; x = a , then f ( 0 ) is

Aicharya · Accepted Answer

F &#8242; a + 2 = &#8747; 0 a f x d x &#8658; F &#8242; &#8242; a = f a ...(i) &#8757; F x = &#8747; x x 2 + &#960; 6 2 cos 2 &#8289; t &#160; d t &#8756; F &#8242; x = 2 cos 2 &#8289; x 2 + &#960; 6 . 2 x - 2 cos 2 &#8289; x F &#8242; &#8242; x = 4 cos 2 &#8289; x 2 + &#960; 6 - 8 x cos &#8289; x 2 + &#960; 6 . sin &#8289; x 2 + &#960; 6 2 x + 4 cos &#8289; x sin &#8289; x f 0 = F &#8242; &#8242; 0 = 4 &#215; 3 4 = 3

Let F x =   ∫ x x 2 + π 6 2 cos 2 ⁡ t   d t for all x   ∈ R   a n…

Solution

Let F x = &#160; &#8747; x x 2 + &#960; 6 2 cos 2 &#8289; t &#160; d t for all x &#160; &#8712; R &#160; a n…

Solution

Let F x = ∫ x x 2 + π 6 2 cos 2 ⁡ t d t for all x ∈ R a n…