计算定积分 ∫(1／α,α)arctanx／x dx（α＞1）

elim

题：计算\(\,\displaystyle\int_{1/\alpha}^\alpha\frac{\arctan x}{x}dx\;(\alpha>1)\)

elim

题：计算\(\,\displaystyle\int_{1/\alpha}^\alpha\frac{\arctan x}{x}dx\;(\alpha>1)\)
解：\(\,\displaystyle\int_{1/\alpha}^\alpha\frac{\arctan x}{x}dx\overset{t=1/x}{=}\int_{1/\alpha}^{\alpha}\frac{\arctan\frac{1}{t}}{t}dt,\;\;\arctan x+\arctan\frac{1}{x}=\frac{\pi}{2}\)
\(\therefore\;\;\displaystyle\int_{1/\alpha}^{\alpha}\frac{\arctan x}{x}dx=\frac{\pi}{4}\int_{1/\alpha}^{\alpha}\frac{dx}{x}=\frac{\pi}{2}\ln\alpha.\)

luyuanhong

楼上 elim 的帖子很好！已收藏。