Question

$\int \frac{cosx-sinx}{cosx+sinx}.(2+2sin2x)dx=$

Answer 1

$\begin{array}{c}\int \frac{\cos x-\sin x}{\cos x+\sin x}\cdot \left(2+2{\sin }^{2}xdx\right)\\ \Rightarrow 2\times \int \frac{\cos x-\sin x}{\left(\cos x+\sin x\right)}\left({\sin }^{2}x+{\cos }^{2}x{\sin }^{2}x\right)dx\\ \Rightarrow 2\int {\frac{\cos x-\sin x}{\cos x+\sin x}\times \left[\sin x+\cos x\right]}^{2}dx\\ \Rightarrow 2\int \left(\cos x-\sin x\right)\left(cosx+\sin x\right)dx\\ \Rightarrow 2\int {\cos }^{2}x-{\sin }^{2}xdx\\ \Rightarrow 2\times \int {\cos }^{2}xdx\\ \Rightarrow 2\times \frac{{\sin }^{2}x}{2}+c\\ \Rightarrow {\sin }^{2}x+cAns.\end{array}$