求广义积分上限是正无穷,下限是0,被积函数是(1+x^2)/(1+x^4)提示,做变换u=x-1/x答案是pi/根号2

Question

求广义积分
上限是正无穷,下限是0,被积函数是(1+x^2)/(1+x^4)
提示,做变换u=x-1/x
答案是pi/根号2

Answer 1

令u=x-1/x,则u^2=x^2+1/x^2-2 √2表示根号2
∫(1+x^2)/(1+x^4)dx
=∫(1+1/x^2)/(x^2+1/x^2)dx
=∫d(x-1/x)/(x^2+1/x^2-2)+2
=∫du/u^2+2
=1/√2arctan(u/√2)+C
x取正无穷时,u取+∞,x取0时,u取-∞,代入即得
(1/√2)*(π/2)-[(1/√2)*(-π/2)]=π/√2