If ∫ 0 π 3 cos 4 x d x = a π + b 3 , where a and b are rational numbers, then 9 a + 8 b is equal to:

Step-by-step solution

Let, I = ∫ 0 π 3 cos 4 x d x ⇒ I = 1 4 ∫ 0 π 3 2 cos 2 x 2 d x ⇒ I = 1 4 ∫ 0 π 3 1 + cos 2 x 2 d x ⇒ I = 1 4 ∫ 0 π 3 1 + cos 2 2 x + 2 cos 2 x d x ⇒ I = 1 4 ∫ 0 π 3 1 + 2 cos 2 x d x + 1 8 ∫ 0 π 3 2 cos 2 2 x d x ⇒ I = 1 4 ∫ 0 π 3 1 + 2 cos 2 x d x + 1 8 ∫ 0 π 3 1 + cos 4 x d x ⇒ I = 1 4 x + sin 2 x + 1 8 x + sin 4 x 4 0 π 3 ⇒ I = 1 4 π 3 + sin 2 π 3 + 1 8 π 3 + sin 4 π 3 4 ⇒ I = π 12 + 3 8 + π 24 - 3 64 ⇒ I = π 8 + 7 3 64 So, on comparing with given value we get, a = 1 8 , b = 7 64 ⇒ 9 a + 8 b = 9 8 + 7 8 ⇒ 9 a +