The values of ∫ - π 2 π 2 sin 2 ⁡ x 1 + 2 x d x is

Question

The values of &#8747; - &#960; 2 &#960; 2 sin 2 &#8289; x 1 + 2 x d x is

Aicharya · Accepted Answer

I = &#160; &#8747; - &#960; 2 &#960; 2 sin 2 x 1 + 2 x &#160; d x &#160; &#160; &#160; . . . ( i ) As &#8747; a b f ( x ) d x = &#8747; a b f ( a + b &#8722; x ) d x I = &#8747; - &#960; 2 &#960; 2 sin 2 &#8289; - x 1 + 2 - x d x I = &#160; &#8747; - &#960; 2 &#960; 2 2 x sin 2 &#8289; x 1 + 2 x d x &#160; &#160; &#160; &#160; . . . ( ii ) By adding Equations ( i ) and ( ii ) , 2 I = &#160; &#8747; - &#960; 2 &#960; 2 1 + 2 x sin 2 &#8289; x 1 + 2 x d x 2 I = 2 &#8747; 0 &#960; 2 sin 2 &#8289; x d x I = &#8747; 0 &#960; 2 cos 2 &#8289; x d x 2 I = &#160; &#8747; 0 &#960; 2 sin 2 &#8289; x + cos 2 &#8289; x d x 2 I = &#160; &#8747; 0 &#960; 2 1 &#160; d x I = &#960; 2 &#215; 1 2 = &#960; 4

The values of &#8747; - &#960; 2 &#960; 2 sin 2 &#8289; x 1 + 2 x d x is

Solution

The values of ∫ - π 2 π 2 sin 2 ⁡ x 1 + 2 x d x is