Question

Integrating (x+sinx)/(1+cosx) this function. The answer is x*tan(x/2)+c but i need to know how to get that answer. Since answer contains tan(x/2) i thought we might use trig substitution but i wasnt able to do that.

Answer 1

I found some help in my calculus book but this would take ages to do here...

Answer 2

any tips you can give could help me coz i know all the techniques but not sure which ones i need to use :S.

Answer 3

Im not even 100% that this would help you.... Anyway I will try my best.
So you can use the substitution
\[z=\tan {x/2} \]

Answer 4

By trigonometric relation you have
\[\tan {x/2}=\frac{ \sin x }{ {1+\cos x} }\]

Answer 5

Wow it is getting tricky :D
I can prove the above you need it, should I?

Answer 6

Now the next step is again trigonometry, a lot.
The results are;
\[\cos x=\frac{ {1-z^2} }{ {1+z^2} }\]
and
\[\sin x= \frac{ {2z} }{{1+z^2} }\]

Answer 7

Where z is tan(x/2)

Answer 8

i know but x=2tan^-1(z) doesnt it

Answer 9

This brings you to
\[dx=\frac{ 2dz }{ {1+z^2} }\]

Answer 10

Yes

Answer 11

I missed out a lot of steps but this is nearly a full page in my book..... (Thomas`Calculus)

Answer 12

i converted the equation in terms of z but it looked so complicated and when i saw the result i thought maybe there is something easier. Coz the derivative of numerator is the denominator. but it is very hard with trig subs. i guess

Answer 13

To be honest I am glad that I do not have to do these anymore

Answer 14

Thanks a lot for your help anyway. My final is tomorrow so hopefully i wont have to do these ever again :D