Evaluate the integral. (use c for the constant of integration.) ∫2tan⁴(x) sec⁶(x) dx.

Solution:

Given the integral: ∫2tan⁴(x) sec⁶(x) dx

⇒ 2 ∫tan⁴(x) sec⁴(x) sec²(x) dx

⇒ 2 ∫tan⁴(x) (sec²(x))² sec²(x) dx

⇒ 2 ∫tan⁴(x) (tan²x + 1)² sec²(x) dx

Let tan x = t ⇒ sec²x dx = dt

⇒ 2 ∫t⁴ (t² + 1)² dt

⇒ 2 ∫t⁴ (t⁴ + 2t² + 1) dt 

⇒ 2 ∫(t⁸ + 2t⁶ + t⁴) dt

⇒ 2 [t⁹ / 9 + 2 × t⁷ / 7 + t⁵ / 5] + C

⇒ 2/9 (tan x)⁹ + 2/7 (tan x)⁷ + 2/5 (tan x)⁵ + C

Evaluate the integral. (use c for the constant of integration.) ∫2tan⁴(x) sec⁶(x) dx.

Summary:

Integration of ∫2tan⁴(x) sec⁶(x) dx is 2/9 (tan x)⁹ + 2/7 (tan x)⁷ + 2/5 (tan x)⁵ + C