Integrate (sec x) / (1+sin x)

Hi there, I have a problem:

How do I integrate:
(sec x) / (1+sin x)?
I know that (sec x) = 1/ (cos x)
therefore, I`d get 1 / [(cos x)(1+sin x)]
1/ [cos x + sinxcosx]
I know the trig. identity: sinxcosx = 1/2 (sin x)^2 but I have no clue what to do with it. I`ve been trying to substitute in my head but nothing seems to work. Am I doing something wrong? I`ve been busting my head on this problem for 30 minutes...

Any help would be very appreciated.

actually, sinxcosx = (1/2)sin2x

no ^2

You are on the right track.

(secx)(1-sinx) /(1+sinx)(1-sinx)

(secx - tanx) / (1 - sin^2(x))

(secx - tanx) / (cos^2)

(secx / cos^2x) - (tanx / cos^2x)

(sec^3(x)) - (sin/cos^3(x))

Now we are ready to integrate.

Start with left hand side:
u = secx
du = secxtanx
dv = sec^2x
v = tanx
(secxtanx) - ∫ (secxtanx)tanx dx

secxtanx - ∫ secx(sec^2x - 1) dx

secx tanx - ∫ sec^3x - secx

secxtanx - ln(tanx + secx) - ∫sec^3x

but sec^3 is what we want to integrate in the first place so:
I = secxtanx - ln(tanx + secx) - I

2I = sectanx - ln(tanx + secx)

I = (1/2)(secxtanx) - (1/2)(ln(tanx+secx))

Thats for the left hand side ONLY...

lets move on to the right...
sin/cos^3

u = cosx
du = -sinsx
-∫ du / u^3
-∫ u^-3 du
(1/2)(u^-2) + C
(1/2cos^2(x)) + C

Slap it all together:
(1/2)(secxtanx) - (1/2)(ln(tanx+secx)) - (1/2cos^2(x)) + C

∫ secx/(1+sinx) dx

∫ secx(1-sinx) / (1-sin²x) dx

∫ secx(1-sinx) / cos²x dx

∫ sec³x - tanx sec²x dx

= ∫ secx sec²x dx - ∫ tanx sec²x dx

put, tanx = t
sec²x dx = dt

= ∫ √(1+t²) - t dt

we know, integration of √(a²+t²) = t/2 √(a²+t²) + a²/2 ln|t+√(a²+t²)|

apply here,

= t/2 √(1+t²) + 1/2 ln|t+√(1+t²)| - t²/2 + c

where, t = tanx

= (tanx)/2 √(1+tan²x) + 1/2 ln|tanx+√(1+tan²x)| - (tan²x)/2 + c

= (secx tanx)/2 + 1/2 ln|tanx + secx| - (tan²x)/2 + c

∫(sec x)/(1 + sin x) dx = ∫1/(cos x)(1 + sin x) dx

= ∫(1 - sin x)/(cos x)(1 - sin² x) dx = ∫(1 - sin x)/cos^3(x) dx = ∫sec^3(x) - sin(x)/cos^3(x) dx

∫sec^3(x) dx - ∫sin(x)/cos^3(x) dx

Use integration by parts for the first one and a substitution of z = cos(x) for the second integral.

EDIT:

It became cos^3(x) because cos(x)(1 - sin² x) = cos(x)(cos²(x)) = cos^3(x)

See my pic for pure algebra and trig manipulation, no need for integrate by parts or substitution
http://p13.freep.cn/p.aspx?u=v20_p13_pho…