Question

integral(1/(1+cosx))
step by step please

Answer 1

for some reason I'm thinking of multiplying by bottom's conjugate on top and bottom first

Answer 2

This should give you a pretty easy form to integrate afterwards
of course you might want to write the resulting fraction as two fractions

Answer 3

that was my instinct also but it needs some fancy u sub after that in order work i think

Answer 4

fancy sub not needed

Answer 5

reg sub needed :p

Answer 6

itll look like integral((cscx)^2 -csc(x)cot(x))

Answer 7

then what

Answer 8

well the derivative of cot(x) is -csc^2(x)
and the derivative of csc(x) is -csc(x)cot(x)
isn't ?

Answer 9

isn't it?*

Answer 10

im lost. so what would be the next step?

Answer 11

if you know the derivative of cot(x) is -csc^2(x)
then you know the integral of -csc^2(x) is cot(x)

Answer 12

if you know the integral of -csc^2(x) is cot(x)
then you know integral of csc^2(x) is -cot(x)

Answer 13

well of course +c

Answer 14

oh ya good point

Answer 15

problem is wolfram says: tan(x/2)

Answer 16

how do i convert it to that?

Answer 17

so you see we have
\[-\cot(x)+\csc(x)+C\]
now:

let's see tan(x/2)...
\[\tan(x/2)=\frac{\sin(x)}{1+\cos(x)}\]

What would happen if you wrote your answer in terms of sin and cos
and combine the fractions

Answer 18

(1-cosx)/sinx - no?

Answer 19

oh got it

Answer 20

that can be multiplied w. conjugate to sinx/(1+cosx) thanks

Answer 21

thanks

Answer 22

great stuff

Answer 23

any other ways of doing this problem that might be quicker?

Answer 24

well you weren't asked to show it was tan(x/2)+C were you
just wanted to see if you got the same as wolfram right?

Answer 25

ya

Answer 26

Off the top of my head I can't think of a shorter way
but actually I remember for some trig expressions
or when integrating some trig expressions
you might want to consider trig substitutions like u=tan(x/2)
I think there might be another one
let me see if I can find it

Answer 27

http://www.math.ucsd.edu/~ebender/20B/trig.pdf
I wouldn't exactly call it shorter though

Answer 28

ok cool thanks