Expert Integration Help: Simplifying \frac{1}{(1+sinx)} in Minutes

[gona]

Integrating \frac{1}{(1+sinx)}

i just started learning Integration last week so not exactly sure how to approch this type of question.

 

[daveb]

Do you know how to integrate 1/sinx?

 

[gona]

no this is the first time i have seen where a trigonometric function is on the denominator

but now that i think about it

sinx = \sqrt{1-cos^2x}
and arcsin was equal to \frac{1}{((1-x^2)}

so i guess i could use the U substituition method
and then the answer would be...arcsin(cosx)? I am not too sure

 

[Dick]

No, no arcsin. But you might want to try multiplying numerator and denominator by (1-sin(x)). It may look more familiar.

 

[daniel_i_l]

You could also try the subsitution u = tan(x/2). It looks a little messy but everything cancels out.

 

[gona]

well I am not sure if I am doing the right thing but here goes...

sinx = \frac{2tan(\frac{x}{2})}{1+tan^{2}(\frac{x}{2})}

soo then i symplify the equation so that it is

\frac{1+tan^{2}(\frac{x}{2})}{1+tan^{2}(\frac{x}{2})+2tan(\frac{x}{2})}

the i used u = tan(\frac{x}{2})

so i get \frac{1+u^{2}}{(u+1)^{2}}

What should i do from here or is this the right way at all?

 

[Dick]

It looks to me like you are taking the long way around. Try multiplying numerator and denominator of your original problem by (1-sin(x)). You get (1-sinx)/cos^2(x). If you split that into two integrals you shouldn't have any problem with either of them.

 

[daniel_i_l]

Dick solution is quicker in this case but to integrate things like
1/(1+cosx+sinx) the substitution u=tan(x/2) is good.
But notice that it isn't x = tan(u/2) but rather u=tan(x/2). In order to get this into something the the example I gave you can show with so trig identities that if u=tan(x/2) then:
dx = 2du/(1+u^2)
sinx = 2u/(1+u^2)
cosx = (1-u^2)/(1+u^2)
If you substitue all that you the (1+u^2)s candel out and you get some rational function which you can solve with the typical rational function method (breaking it into elementry functions...)

 

[gona]

wow by the way i got the answer using what dick said its actually pretty easy once u break it up...now I am going to try Daniels question going to see if i can get those now :) thnx a lot for the help by the way
the answe was tanx - secx +c