How to integrate tan(x)sec^3(x)?

To integrate tan(x)sec^3(x), use substitution with u = sec(x) + tan(x).

To integrate tan(x)sec^3(x), we can use substitution with u = sec(x) + tan(x). This is because the derivative of sec(x) + tan(x) is sec(x)tan(x) + sec^2(x), which is similar to the integrand.

First, we need to find du/dx. Using the chain rule, we have:

du/dx = d/dx(sec(x) + tan(x))
= sec(x)tan(x) + sec^2(x)

Now, we can rewrite the integrand in terms of u:

tan(x)sec^3(x) = (sec^2(x) - 1)sec(x)tan(x)
= (u^2 - 1)du/dx

Substituting u and du/dx into the integral, we get:

∫tan(x)sec^3(x) dx = ∫(u^2 - 1)du
= (u^3/3 - u) + C
= (sec^3(x) + 3sec(x) - tan(x)) / 3 + C

Therefore, the integral of tan(x)sec^3(x) is (sec^3(x) + 3sec(x) - tan(x)) / 3 + C.