\(\int \frac{d x}{1+\sin x}=\int \frac{d x}{1+\sin x} \times \frac{1-\sin x}{1-\sin x}=\int \frac{(1-\sin x)}{1-\sin ^{2} x} d x\)
\(=\int \frac{1-\sin x}{\cos ^{2} x} d x=\int\left(\frac{1}{\cos x}-\frac{\sin x}{\cos x \cos x}\right)\)
\(=\int\left(\sec ^{2} x-\tan x \sec x\right) d x\)
\(=\int \sec ^{2} x d x-\int \tan \cdot \sec x d x=\tan x-\sec x+c\)
