Can anyone explain how to find the integral of sec(x) or csc(x) with respect to x?

Question

anonymous · Answer

lalaly is the integral master today

lalaly · Answer

$$\int\limits{secxdx}=\int\limits{secx \times \frac{secx+tanx}{secx+tanx}}dx$$$$=\int\limits{\frac{\sec^2 x+secxtanx}{secx+tanx}dx}$$
let u=secx+tanx
so the integration becomes $$\int\limits{\frac{du}{u}}=\ln|u|+C$$$$=\ln|secx+tanx|+C$$

lalaly · Answer

same for csc(x) but u multiply by (cscx+cotx)/(cscx+cotx)

lalaly · Answer

lol satelite:)

inkyvoyd · Answer

Holy crap, no wonder I couldn't figure it out on my own. Are there any other cases where we do this?

anonymous · Answer

lalaly is correct

anonymous · Answer

here is another version. apparently this was a big deal in the 17th century http://en.wikipedia.org/wiki/Integral_of_the_secant_function