If I = 2 π ∫ - π 4 π 4 d x 1 + e s i n x 2 - c o s 2 x then find 27 I 2 equals____

Question

If I = 2 &#960; &#8747; - &#960; 4 &#960; 4 d x 1 + e s i n x 2 - c o s 2 x then find 27 I 2 equals____

Aicharya · Accepted Answer

I = 2 &#960; &#8747; - &#960; 4 &#960; 4 1 1 + e s i n x 2 - c o s 2 x &#8901; d x Using &#8747; - a a f x &#8901; d x = &#8747; 0 a f x + f - x &#8901; d x I = 2 &#960; &#8747; 0 &#960; 4 1 1 + e s i n x 2 - c o s 2 x + 1 1 + e - s i n x 2 - c o s 2 x &#8901; d x I = 2 &#960; &#8747; 0 &#960; 4 1 1 + e s i n x 2 - c o s 2 x + e s i n x e s i n x + 1 2 - c o s 2 x &#8901; d x I = 2 &#960; &#8747; 0 &#960; 4 d x 2 - c o s 2 x I = 2 &#960; &#8747; 0 &#960; 4 d x 2 - 1 - t a n 2 x 1 + t a n 2 x = 2 &#960; &#8747; 0 &#960; 4 1 + t a n 2 x d x 1 + 3 t a n 2 x I = 2 &#960; &#8747; 0 &#960; 4 s e c 2 x d x 1 + 3 t a n 2 x Put t a n &#160; x = t sec 2 &#8289; x d x = d t &#160; and when x = 0 &#8658; t = 0 x = &#960; 4 &#8658; t = 1 I = 2 &#960; &#8747; 0 1 d t 1 + 3 t 2 I = 2 &#960; &#8747; 0 1 d t 1 + 3 t 2 I = 2 &#960; 1 3 t a n - 1 3 t 0 1 &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#8757; &#8201;&#8201; &#8747; 0 1 d x a 2 + x 2 = 1 a tan - 1 x a + C I = 2 3 &#960; t a n - 1 3 - t a n - 1 0 I = 2 3 &#960; &#215; &#960; 3 I = 2 3 3 27 I 2 = 27 &#215; 4 27 = 4

If I = 2 &#960; &#8747; - &#960; 4 &#960; 4 d x 1 + e s i n x 2 - c o s 2 x then find 27 I 2 equals____

Solution

If I = 2 π ∫ - π 4 π 4 d x 1 + e s i n x 2 - c o s 2 x then find 27 I 2 equals____