Question

$\int \frac{\sin x}{1+\sin x}dx$

Answer 1

$$\begin{gathered} \int \frac{\sin x}{1+\sin x}dx \\ \mathrm{Rationalising}\mathrm{the}\mathrm{denominator} \\ \Rightarrow \int \frac{\sin x}{1+\sin x}\times \frac{1-\sin x}{1-\sin x}dx \\ \Rightarrow \int \left(\frac{\sin x-{\sin }^{2}x}{1-{\sin }^{2}x}\right)dx \\ \Rightarrow \int \left(\frac{\sin x}{{\cos }^{2}x}-{\tan }^{2}x\right)dx \\ \Rightarrow \int \left\{\frac{\sin x}{\cos x}\times \frac{1}{\cos x}-\left({\sec }^{2}x-1\right)\right\}dx \\ \Rightarrow \int \left(\sec x\tan x-{\sec }^{2}x+1\right)dx \\ \Rightarrow \sec x-\tan x+x+C \end{gathered}$$