How to integrate tan^4(x)?

To integrate tan^4(x), use the identity tan^2(x) = sec^2(x) - 1 and substitution.

First, rewrite tan^4(x) as (tan^2(x))^2. Then, use the identity tan^2(x) = sec^2(x) - 1 to get:

tan^4(x) = (sec^2(x) - 1)^2

Expand the square to get:

tan^4(x) = sec^4(x) - 2sec^2(x) + 1

Now, substitute u = sec(x) and du/dx = sec(x)tan(x) dx. This gives:

∫tan^4(x) dx = ∫(sec^4(x) - 2sec^2(x) + 1) dx
= ∫(u^4 - 2u^2 + 1) du/(u tan(x))

Simplify the integrand by dividing through by u^3 tan(x):

∫tan^4(x) dx = ∫(u - 2/u + 1/u^3) du

Integrate each term separately:

∫tan^4(x) dx = (1/2)u^2 - 2ln|u| - (1/2u^2) + C

Substitute back in for u:

∫tan^4(x) dx = (1/2)sec^2(x)^2 - 2ln|sec(x)| - (1/2)sec^(-2)(x) + C

Simplify the answer by using the identity sec^2(x) = 1 + tan^2(x):

∫tan^4(x) dx = (1/2)(tan^2(x) + 1)^2 - 2ln|sec(x)| - (1/2)(1 + tan^2(x))^(-1) + C