Question

高等数学积分.1／sin（θ＋pai／4） 从零到pai／2的定积分.

Answer 1

设t=tan(θ/2),则dt=sec(θ/2)*dθ/2=(1+t^2)dθ/2,

sinθ=2t/(1+t^2),cosθ=(1-t^2)/(1+t^2),

∴∫dθ/sin(θ+π/4)

=2√2∫dt/(1+2t-t^2)

=2√2∫du/(2-u^2)

=ln[(√2+u)/(√2-u)]|

=2ln(√2+1）,

其中u=t-1.