Question

∫1/(2+cos x) dx 定积分?

Answer 1

设t=tan(x/2)

则cosx=[cos²(x/2)-sin²(x/2)]/[cos²(x/2)+sin²(x/2)]

=[1-tan²(x/2)]/[1+tan²(x/2)]

=(1-t²)/(1+t²)

dx=d(2arctant)=2dt/(1+t²)

故∫1/(2+cosx)dx=∫1/[2+(1-t²)/(1+t²)]*[2dt/(1+t²)]

=∫2dt/(3+t²)

=2/√3∫d(t/√3)/[1+(t/√3)²]

=2/√3arctan(t/√3)+C