求tan^4(x) ＊sec(x) 的不定积分

求tan^4(x) ＊sec(x) 的不定积分

公式：
J_(n) = ∫ sec^n(x) dx
J_(n) = [sec^(n - 1)(x) sin(x)]/(n - 1) + (n - 2)/(n - 1) J_(n - 2)
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∫ tan^4(x) secx dx
= ∫ (sec²x - 1)² secx dx
= ∫ (sec^4(x) - 2sec²x + 1) secx dx
= ∫ sec^5(x) dx - 2∫ sec³x dx + ∫ secx dx
= J_(5) - 2J_(3) + J
= (1/4) sec^4(x) sinx + 3/4 J_(3)
= (1/4) sec^4(x) sinx - (5/4)J_(3) + J
= (1/4) sec^4(x) sinx - (5/4)[(1/2) sec²x sinx + 1/2 J] + J
= (1/4) sec^4(x) sinx - (5/8) sec²x sinx + (3/8)J
= (1/4) sec^4(x) sinx - (5/8) sec²x sinx + (3/8)ln|secx + tanx| + C
再问： 敢不敢不用那个坑爹的reduction formula...0.0
再答： 呵呵，就慢慢积分吧。 J = ∫ sec^5(x) dx = ∫ sec³x d(tanx) = sec³x tanx - ∫ tanx 3sec²x secx tanx dx = sec³x tanx - 3∫ sec³x tan²x dx = sec³x tanx - 3∫ sec³x (sec²x - 1) dx = sec³x tanx - 3J + 3∫ sec³x dx J = 1/4 sec³x tanx + 3/4 ∫ sec³x dx K = ∫ sec³x dx = ∫ secx dtanx = secx tanx - ∫ tanx secx tanx dx = secx tanx - ∫ secx (sec²x - 1) dx = secx tanx - K + ∫ secx dx K = 1/2 secx tanx + 1/2 ln|secx + tanx| 原式 = 1/4 sec³x tanx + 3/4 [1/2 secx tanx + 1/2 ln|secx + tanx|] - [secx tanx + ln|secx + tanx|] + ln|secx + tanx| + C = (1/8) secx tanx (2sec²x - 5) + (3/8) ln|secx + tanx| + C