计算∫e2x（tanx+1）2dx．

Question

计算∫e^2x（tanx+1）²dx．

Answer 1

原式=(1)/(2)∫(1+tanx)2de2x＝(1)/(2)e2x(1+tanx)2−(1)/(2)∫e2xd(1+tanx)2
=(1)/(2)e2x(1+tanx)2−∫e2x(tanx+1)sec2xdx
=(1)/(2)e2x(1+tanx)2−∫e2xtanxsec2xdx−∫e2xsec2xdx       
=(1)/(2)e2x(1+tanx)2−∫e2xd((1)/(2)tan2x)−∫e2xsec2xdx
=(1)/(2)e2x(1+tanx)2−(1)/(2)e2xtan2x+∫e2xtan2xdx−∫e2xsec2xdx
=(1)/(2)e2x(1+2tanx)−∫e2xdx
=(1)/(2)e2x(1+2tanx)−(1)/(2)e2x+C
=e^2xtanx+C