∫dx／x*√a²－x² 您给看看

∫dx／x*√a²－x² 您给看看
∫dx／x*√a²－x² 您给看看

∫dx／x*√a²－x² 您给看看
令x = a * sinz,dx = a * cosz dz
∫ dx/[x√(a² - x²)]
= ∫ 1/(asinz * |acosz|) * (acosz dz)
= (1/a)∫ cscz dz
= (1/a)ln|cscz - cotz| + C
= (1/a)ln|a/x - √(a² - x²)/x| + C
= (1/a)ln|[a - √(a² - x²)]/x| + C

用换元法
令x=acost
dx=-asintdt
∫dx／(x*√a²－x² )
=∫1/[acost*asint]*(-asint)dt
=-∫1/(acost)dt
=-1/a∫cost/(cos^2t)dt
=-1/a∫1/(1-sin^2t)dsintt
=-1/(2a)∫[1/(1-sint)+1/(1+sint)]dsint
=-1/(2a)[-ln(1-sint)+ln(1+sint)]+C