不定积分∫[dx/(2x^2+1)(x^2+1)^(1/2)]

题目

不定积分∫[dx/(2x^2+1)(x^2+1)^(1/2)]

答案

令x=tan(t) dx=sec^2(t) (x^2+1)^(1/2)=sect2x^2+1=2(x^2+1)-1=2sec^2(t)-1sect=1/cost原式=∫d(sint)/[2-cos^2(t)]=∫d(sint)/[1+sin^2(t)]即∫ds/1+s^2原式=arctans=arctan(sint)