Integration

(1- sin x)/(1- sin x) = 1

We multiply the integrand by (1- sin x)/(1- sin x)

dx (1- sin x) 1 – sin x
-------------- ------------------ = --------------------- dx
(1 + sin x) (1- sin x) 1 – sin^2 (x)

1 – sin z 1 sin x
---------------- dx = ( ------------ - --------------) dx
cos^2 (x) cos^2(x) cos^2(x)

sin x tan x
( sex^2(x) - ----------------) dx = sec^2(x) - -------------- dx
cos x cos x cos x

Integrate [sec^2(x) – tan x sec x ] dx

= tan x – sec x + C

Over the range from x = 0 to x = pi

[tan (pi) – sec (pi) +C] - [tan (0) – sec (0) +C]

tan (pi) – sec (pi) - tan (0) + sec (0)

0 – (-1) – 0 + 1 = +1 + 1 = 2

Integrate 1/(1 + sin x) dx from x = 0 to x = pi = 2 (answer)

You have to know tan x = sin x/ cos x, sin^2 (x) + cos^2 (x) = 1

d/dx tan x = sec^2(x) and d/dx sec x = sec x tan x
Integration of sec^2(x) = tan x + C
Integration of sec x tan x = sec x + C

2012-04-04 11:10:50 補充：
Correction:

Integrate 1/(1 + sin x) dx from x = 0 to x = pi is 2 (answer)