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

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

To integrate sec(x)tan(x), we can use u-substitution. Let u = sec(x) + tan(x), then du/dx = sec(x)tan(x) + sec^2(x). Rearranging, we have sec(x)tan(x) = du/dx - sec^2(x). Substituting this into the integral, we get:

∫sec(x)tan(x) dx = ∫(du/dx - sec^2(x)) dx

The first term can be integrated easily as it is just u with respect to x. For the second term, we can use the identity 1 + tan^2(x) = sec^2(x) to rewrite it as:

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

The second integral can be integrated using the substitution u = tan(x), du/dx = sec^2(x), giving:

∫dx/(1 + tan^2(x)) = ∫du/u^2 + 1 = arctan(u) + C = arctan(tan(x)) + C

Putting everything together, we get:

∫sec(x)tan(x) dx = ln|sec(x) + tan(x)| - arctan(tan(x)) + C

Simplifying the arctan(tan(x)) term, we get:

∫sec(x)tan(x) dx = ln|sec(x) + tan(x)| - x + C