Question

∮L(exsiny-2y)dx+(excosy-2)dy,其中L为上半圆周,（X-a）2+y2=a,y≥0,沿逆时针方向

Answer 1

∫sinx*（1 cosx^2)^0.5dx

=∫sinx*√（1 cosx^2)dx

=∫2sin(x/2)*cos(x/2)*√2[cos(x/2)]^2dx

=2√2∫sin(x/2)*[cos(x/2)]^2dx

=4√2∫sin(x/2)*[cos(x/2)]^2d(x/2)

=-4√2∫[cos(x/2)]^2d[cos(x/2)]

=-4√2 [cos(x/2)]^3 /3 C

(注:第三个等号因为x是0到π,所以x/2是0到π/2,所以可以根号开出来不加绝对值)

定积分就自己做