Question

Evaluate the integral using substitution method.
∫sec(x/3)dx

Answer 1

hint: the integral of sec(u) is

= ln(sec(u) + tan(u))

so now you can guess what U should be :)

Answer 2

oh ok thank you ! =)

Answer 3

here if you want to see all the work:

u = x/3
du = dx

= integral [ sec(u) du ]

this part is fun, my mind was blown when I found out you could do this. It's almost like magic:


= integral [ sec(u) * [sec(u) + tan(u) / sec(u) + tan(u)] du]

= integral [ sec^2(u) + sec(u)tan(u) / sec(u) + tan(u) du]

now lets use a W substitution

w = sec(u) + tan(u)
dw = sec(u)tan(u) + sec^2(u) du

you see how we have a sec^2(u) + sec(u)tan(u) at the top already?

= integral [ (1/w) dw]

= ln(w)

= ln( sec(u) + tan(u))

= ln(sec(x/3) + tan(x/3))

tada!

Answer 4

oh sorry there should be a 1/3 in the front

Answer 5

yes i saw that. thank you very much again