Let f x = 7 tan 8 ⁡ x + 7 tan 6 ⁡ x - 3 tan 4 ⁡ x - 3 tan 2 ⁡ x for all x ∈ - π 2 , π 2 . Then the correct expression(S) is(are)

Step-by-step solution

f x = 7 tan 8 ⁡ x + 7 tan 6 ⁡ x - 3 tan 4 ⁡ x - 3 tan 2 ⁡ x f x = sec 2 ⁡ x 7 tan 6 ⁡ x - 3 tan 2 ⁡ x Now, ∫ 0 π 4 sec 2 ⁡ x 7 tan 6 ⁡ x - 3 tan 2 ⁡ x d x = ∫ 0 1 7 t 6 - 3 t 2 d t = 7 t 7 7 - 3 t 3 3 = 1 7 - 1 3 = 0 Also ∫ 0 π 4 x sec 2 ⁡ x 7 tan 6 ⁡ x - 3 tan 2 ⁡ x d x tan ⁡ x = t ∫ 0 1 tan - 1 ⁡ t 7 t 6 - 3 t 2 d t tan - 1 ⁡ t t 7 - t 3 | 0 1 - ∫ 0 1 1 1 + t 2 t 7 - t 3 d t = - ∫ 0 1 1 1 + t 2 t 3 t 4 - 1 d t = + ∫ 0 1 t 3 1 + t 2 1 - t 2 d t 1 + t 2 = ∫ 0 1 t 3 - t 5 d t = 1 4 - 1 6 = 1 12