第1个回答 2017-12-18
令x = 3secθ,dx = 3secθtanθ dθ
√(x² - 9) =√(9sec²θ - 9) = 3|tanθ|
若x > 3,|tanθ| = tanθ
若x < - 3,|tanθ| = - tanθ
原式 = ∫ 3|tanθ|/[3secθ] * [3secθtanθ dθ]
= 3∫ tan²θ dθ
= 3∫ (sec²θ - 1) dθ
= 3tanθ - 3θ + C
= 3[√(x² - 9)/3] - 3[arcsec(x/3)] + C
= √(x² - 9) - 3arccos(3/x) + C
综合:
∫ √(x² - 9)/x dx = { √(x² - 9) - 3arccos(3/x) + C,若x > 3
{ - √(x² - 9) + 3arccos(3/x) + C,若x < - 3