How to integrate sec^4(x)?

To integrate sec^4(x), use the substitution u = tan(x) and simplify using trigonometric identities.

First, rewrite sec^4(x) as (sec^2(x))^2. Then, using the identity 1 + tan^2(x) = sec^2(x), substitute u = tan(x) and du = sec^2(x) dx. This gives:

∫sec^4(x) dx = ∫(sec^2(x))^2 dx
Let u = tan(x), du = sec^2(x) dx
= ∫(1 + tan^2(x))^2 sec^2(x) dx
= ∫(1 + u^2)^2 du

Expand the integrand and integrate term by term:

= ∫(1 + 2u^2 + u^4) du
= u + 2/3 u^3 + 1/5 u^5 + C

Substitute back u = tan(x) to get the final answer:

∫sec^4(x) dx = tan(x) + 2/3 tan^3(x) + 1/5 tan^5(x) + C