intergration!!!!

Use Weierstrass substitution. I assume you know Weierstrass substitution

Let t = tan (x/2), sin x = 2t/(1 + t^2), cos = (1 – t^2)/(1 + t^2), dx = 2/(1 + t^2)dt
∫1/(sinx + tan x)dx =1/(sinx + sin x/cos x) dx

1
∫------------------------------------------------ dx
2t 2t (1 + t^2)
------------- + ------------- --------------
(1 + t^2) (1 + t^2) (1 – t^2)

1
∫-------------------------------- dx
2t 2t
------------- + -------------
(1 + t^2) (1 – t^2)

1
∫----------------------------------- dx
2t(1 – t^2) + 2t (1 + t^2)
-----------------------------------
(1 + t^2) (1 – t^2)

1
∫------------------------------- dx
2t – 2t^3 + 2t + 2t^3
-------------------------------
(1 + t^2) (1 – t^2)

1
∫-------------------- dx
4t
---------------------
(1 + t^2) (1 – t^2)

(1 + t^2) (1 – t^2) 2
∫------------------------- ------------ dt
4t (1 + t^2)

(1 – t^2)
∫------------- dt
2t

∫ [½ (1/t) – ½(t)] dt
½ ln t – ¼ t^2 + C
½ ln |(tan(x/2)| – ¼ tan^2(x/2) + C , where C is a constant

大學做過, 實在太久了, 如果有錯,希望有人指正!