The value of ∫ 0 2 π x sin 8 ⁡ x sin 8 ⁡ x + cos 8 ⁡ x d x is equal to:

Step-by-step solution

Given, I = ∫ 0 2 π x sin 8 ⁡ x sin 8 ⁡ x + cos 8 ⁡ x d x     . . . i Use the property ∫ a b f x d x = ∫ a b f a + b - x d x we get, I = ∫ 0 2 π 2 π - x sin 8 ⁡ 2 π - x sin 8 ⁡ 2 π - x + cos 8 ⁡ 2 π - x d x = ∫ 0 2 π 2 π - x sin 8 ⁡ x sin 8 ⁡ x + cos 8 ⁡ x d x     . . . i i Adding equation i   &   i i   we get, 2 I = ∫ 0 2 π 2 π sin 8 &#828