Question

Evaluate: ∫ tan³ x .sec⁵ x dx

Answer 1

∫ tan³ x sec⁵ x dx

= ∫ sec⁴ x .tan² x sec x .tan x dx 

= ∫ sec⁴ x (sec² x – 1)sec x .tan x dx 

Let sec x = t

sec x .tan x dx = dt 

= ∫ t⁴ (t² – 1) dt, 

= ∫ (t⁶ – t⁴) dt

= ((t⁷/ 7) – (t⁵/ 5)) + c 

= ((sec⁷ x/ 7)  – (sec⁵ x/ 5)) + c