Let, I n = ∫ tan n x d x n > 1 . If I 4 + I 6 = a tan 5 x + b x 5 + c , then the ordered pair a , b , is equal to

Step-by-step solution

Given,  I n = ∫ tan n x d x ⇒ I 4 + I 6 = ∫ tan 4 x d x + ∫ tan 6 x d x ⇒ I 4 + I 6 = ∫ tan 4 x d x 1 + tan 2 x d x ⇒ I 4 + I 6 = ∫ tan 4 x sec 2 x d x Put, tan x = t ⇒ sec 2 x d x = d t ⇒ I 4 + I 6 = ∫ t 4 d t ⇒ I 4 + I 6 = t 5 5 + c , where c is the constant of integration. ⇒ a tan 5 x + b x 5 + c = tan 5 x 5 + c ∴   a = 1 5 &   b = 0