【正确答案】解：在极坐标系下,积分区域 则 其中 ∫sec3θdθ=∫sec θdtan θ =sec θ·tan θ-tan θdsec θ =sec θ·tan θ-∫tan2θ·sec θdθ =sec θ·tan θ-∫(sec2θ-1)·sec θdθ =sec θ·tan θ-∫sec3θdθ+∫sec θdθ =see θ·tan θ-∫sec3θdθ+ln|sec θ+tan θ|, 即∫sec3θdθ=(sec θ-tan θ+ln|secθ+tan θ|)+C, 所以