Question

$\int \frac{dx}{\cos x-\sin x}$ is equal to-

Answer 1

$\int \frac{dx}{cosx-sinx}$

$\int \frac{dx}{\frac{1-ta{n}^{2}x/1}{1+ta{n}^{2}x/2}-\frac{2tanx/1}{1+ta{n}^{2}x/2}}$

$\int \frac{(1+ta{n}^{2}x/2)dx}{1-ta{n}^{2}x/2-2tanx/2}$

$\int \frac{se{c}^{2}x/2}{1-ta{n}^{2}x/2-2tanx/2}dx$

$tanx/2=t$

$se{c}^{2}x/2dx=2dt$

$2\int \frac{dt}{1-t^2+2t}$

$2\int \frac{dt}{1-(t^2-2t+1)+1}$

$2\frac{dt}{2-(t-1{)}^2}$

$2\int \frac{dt}{(\sqrt{2}{)}^2-(t-1{)}^2}$

$2\frac{1}{\sqrt{2}}ln\left|\frac{\sqrt{2}+t-1}{\sqrt{2}-t+1}\right|+C$

$\frac{1}{\sqrt{2}}ln\left|\frac{\sqrt{2}+tanx/2-1}{\sqrt{2}+1-tanx/2}\right|+C$