If the integral 525 ∫ 0 π 2 sin 2 x cos 11 2 x 1 + cos 5 2 x 1 2 d x is equal to n 2 − 64 , then n is equal to ________

Step-by-step solution

Let, I = ∫ 0 π 2 sin 2 x cos 11 2 x 1 + cos 5 2 x 1 2 d x Let cos x = t 2 ⇒ sin x d x = - 2 t d t ∴ I = - 4 ∫ 1 0 t 2 ⋅ t 11 1 + t 5 t d t ⇒ I = 4 ∫ 0 1 t 14 1 + t 5 d t Now, let 1 + t 5 = k 2 ⇒ 5 t 4 d t = 2 k d k ∴ I = 4 ⋅ ∫ 1 2 k 2 − 1 2 ⋅ k · 2 k 5 d k ⇒ I = 8 5 ∫ 1 2 k 6 − 2 k 4 + k 2 d k ⇒ I = 8 5 k 7 7 − 2 k 5 5 + k 3 3 1 2 ⇒ I = 8 5 8 2 7 − 8 2 5 + 2 2 3 − 1 7 + 2 5 − 1 3 ⇒ I = 8 5 22 2 105 − 8 105 ⇒ 525 ⋅ I = 176 2 − 64 Hence, on comparing we get, n = 176