Question

高数中不定积分的问题，在线等，求详解

Answer 1

令x=sint
dx=costdt
∫√(1-x^2)/(2x^2-1)dx
=∫cos^2t/(2sin^2t-1)dt
=1/2∫(cos2t+1)/(-cos2t)dt
=-1/2∫dt-1/4∫sec2td(2t)
=-1/2t-1/4ln(sec2t+tan2t)+C
=-1/2arcsinx-1/4ln[1/cos2t+2tant/(1-tan^2t)]+C
=-1/2arcsinx-1/4ln[1/(1-2sin^2t)+2tant/(1-tan^2t)]+C
=-1/2arcsinx-1/4ln{[1+2x√(1-x^2)]/(1-2x^2)}+C
=-1/2arcsinx-1/4ln[1+2x√(1-x^2)]+1/4ln(1-2x^2)+C
注：tant=x/√(1-x^2)
2x/√(1-x^2)/[1-x^2/(1-x^2)]
=[1+2x√(1-x^2)]/(1-2x^2)

Answer 2

换元令x=sint。