不定积分∫(x-1)/√(1-4x^2)dx
问题描述：
不定积分∫(x-1)/√(1-4x^2)dx
答
∫(x-1)/√(1-4x^2)dx
= (-1/4) ∫d√(1-4x^2) - ∫1/√(1-4x^2)dx
=(-1/4) √(1-4x^2) - ∫1/√(1-4x^2)dx
let
2x= sina
2dx= cosada
∫1/√(1-4x^2)dx
=(1/4)∫da
= a/4 + C
=arcsin(2x) + C'
∫(x-1)/√(1-4x^2)dx
=(-1/4) √(1-4x^2) - ∫1/√(1-4x^2)dx
=(-1/4) √(1-4x^2) - arcsin(2x) + C
答
因为分母是√1-4x^2所以设 x=cost/2dx=-sint/2 *dt则√1-4x^2=sint所以原式化为∫(cost/2-1)/sint *(-sint/2 *dt)=1/2∫(1-cost/2)dt=1/2∫dt -1/4 * ∫costdt=1/2*t-1/4∫dsint=t/2-sint/4因为x=cost/2t=arccos(2x)...