Evaluate the integral (use c for the constant of integration) ∫8tan⁴(x) sec⁶(x) dx

Solution:

Given the integral: ∫8tan⁴(x) sec⁶(x) dx

⇒ 8 ∫tan⁴(x) sec⁴(x) sec²(x) dx

⇒ 8 ∫tan⁴(x) (sec²(x))2 sec²(x) dx

⇒ 8 ∫tan⁴(x) (tan²x + 1)² sec²(x) dx

Let tan x = t ⇒ sec²x dx = dt

⇒ 8 ∫t⁴ (t² + 1)² dt

⇒ 8 ∫t⁴ (t⁴ + 2t² + 1) dt 

⇒ 8∫(t⁸ + 2t⁶+ t⁴) dt

⇒ 8 [t⁹/9 + 2 × t⁷ / 7 + t⁵ /5] + C

⇒ 8/9 (tan x)⁹ +16/7 (tan x)⁷ + 8/5 (tan x)⁵ + C

Evaluate the integral (use c for the constant of integration) ∫8tan⁴(x) sec⁶(x) dx

Summary:

Integration of ∫8tan⁴(x) sec⁶(x) dx is 8/9 (tan x)⁹ +16/7 (tan x)⁷ + 8/5 (tan x)⁵ + C.