Question

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

Answer 1

$$\begin{gathered} \int \frac{\cos x}{1+\cos x}dx \\ =\int \frac{\cos x\left(1-\cos x\right)}{\left(1+\cos x\right)\left(1-\cos x\right)}dx \\ =\int \frac{\cos x-{\cos }^{2}x}{1-{\cos }^{2}x}dx \\ =\int \frac{\cos x-{\cos }^{2}x}{{\sin }^{2}x}dx \\ =\int \frac{\cos x}{{\sin }^{2}x}-\frac{{\cos }^{2}x}{{\sin }^{2}x}dx \\ =\int \left(\cot x\mathrm{cosec}x-{\cot }^{2}x\right)dx \\ =\int \left(\cot x\mathrm{cosec}x-\mathrm{co}{\sec }^{2}x+1\right)dx \\ =\int \cot x\mathrm{cosec}xdx-\int \mathrm{co}{\sec }^{2}xdx+\int 1dx \\ =-\mathrm{cosec}x+\cot x+x+C \end{gathered}$$