求∫(√(1-x²))/x²dx

求∫(√(1-x²))/x²dx
求∫(√(1-x²))/x²dx

求∫(√(1-x²))/x²dx
∫(√(1-x²))/ x² dx
let
x = sina
dx = cosa da
∫(√(1-x²))/ x² dx
=∫ [cosa/(sina)^2] cosa da
=∫ (cota)^2 da
=∫ [(csca)^2-1] da
=-cota -a + C
= -√(1-x²) /x - arcsinx + C

设x=sint dx=cost dt t=arcsinx
则
原式=∫ (cost/sin²t) *cost dt
=∫ cost²/sin²t dt
=∫ cot²t dt
=∫(csc²t-1)dx
=-cot t-t+C
则 原式=-√(1-x²)/x -arcsinx+C