參變量在哪兒?積分區(qū)間是什么?被積函數(shù)正確嗎?已補(bǔ)充∫(0->1)arctanx dx/(x*sqrt(1-x^2))代入arctanx/x= ∫(0->1)dy/(1+x^2*y^2) ＝ ∫(0->1)dx/sqrt(1-x^2)* ∫(0->1)dy/(1+x^2*y^2) 交換積分順序 ＝ ∫(0->1)dy * ∫(0->1)dx/(sqrt(1-x^2)* (1+x^2*y^2))做變量替換x＝sina ＝ ∫(0->1)dy∫(0->pi/2) da/(1+sin^2a*y^2) 分子分母除以cos^2a，再做變量替換x＝tana ＝ ∫(0->1)dy∫(0->無窮)dx/((1+y^2)*x^2+1) ＝ ∫(0->1)dy arctan(sqrt(1+y^2)*x)/sqrt(1+y^2)|上限無窮下限0 ＝ pi/2*∫(0->1))1/sqrt(1+y^2)dy ＝pi/2*ln(y+sqrt(1+y^2))|上下1下限0 ＝pi/2*ln(1+sqrt(2))。