∫ex(1+sin⁡x)1+cos⁡xdx is equal to

∫ex(1+sin⁡x)1+cos⁡xdx=∫ex[1+2sin⁡(x/2)cos⁡(x/2)]2cos2⁡(x/2)dx=∫ex12sec2⁡x2+tan⁡x2dx=12∫exsec2⁡x2dx+∫extan⁡x2dx=12∫exsec2⁡x2dx+extan⁡x2−12∫exsec2⁡x2+C[ integrating by parts ]=extan⁡x2+C