Question

Integrate $\frac{1}{1+sinxcosx}$

Answer 1

$\int \frac{1}{1+sinxcosx}dx\int \frac{1}{si{n}^{2}x+co{s}^{2}x+sinxcosx}dxdividebyco{s}^{2}x\int \frac{se{c}^{2}x}{ta{n}^{2}x+1+tanx}dxputtanx=tse{c}^{2}xdx=dt\int \frac{dt}{t^2+t+1}\int \frac{dt}{(t+\frac{1}{2}{)}^2+\frac{3}{4}}x=t+\frac{1}{2}d=\frac{t^2}{2}=\frac{1}{a}ta{n}^{-1}\frac{x}{a}=\frac{2}{\sqrt{3}}ta{n}^{-1}\left\{\frac{t+\frac{1}{2}}{\sqrt{3}.2}\right\}=\frac{2}{\sqrt{3}}ta{n}^{-1}\left\{\frac{\frac{(2t+1)}{x}}{\frac{3}{x}}\right\}=\frac{2}{\sqrt{3}}ta{n}^{-1}\frac{2tanx+1}{\sqrt{3}}$