这个不定积分怎么求 这个不定积分怎么求
secx\u7684\u4e0d\u5b9a\u79ef\u5206\u600e\u4e48\u6c42\u6709\u597d\u51e0\u79cd\u65b9\u6cd5\u7684\uff1a\u6700\u5e38\u7528\u7684\u662f\u222b secx dx = ln|secx + tanx| + C
\u7b2c\u4e00\u79cd\u6700\u5feb\uff1a
\u222b secx dx
= \u222b secx • (secx + tanx)/(secx + tanx) dx
= \u222b (secxtanx + sec²x)/(secx + tanx) dx
= \u222b d(secx + tanx)/(secx + tanx)
= ln|secx + tanx| + C
\u7b2c\u4e8c\u79cd\uff1a
\u222b secx dx
= \u222b 1/cosx dx = \u222b cosx/cos²x dx = \u222b dsinx/(1 - sin²x)
= (1/2)\u222b [(1 - sinx) + (1 + sinx)]/[(1 - sinx)(1 + sinx)] dsinx
= (1/2)\u222b [1/(1 + sinx) + 1/(1 - sinx)] dsinx
= (1/2)[ln|1 + sinx| - ln|1 - sinx|] + C
= (1/2)ln|(1 + sinx)/(1 - sinx)| + C
= ln| \u221a(1 + sinx)/\u221a(1 - sinx) | + C
= ln| [\u221a(1 + sinx)]²/\u221a[(1 - sinx)(1 + sinx)] | + C
= ln| (1 + sinx)/cosx | + C
= ln|secx + tanx| + C
\u7b2c\u4e09\u79cd\uff1a
\u222b secx dx = \u222b 1/cosx dx
= \u222b 1/sin(x + \u03c0/2) dx,\u6216\u8005\u5316\u4e3a1/sin(\u03c0/2 - x)
= \u222b 1/[2sin(x/2 + \u03c0/4)cos(x/2 + \u03c0/4)] dx,\u5206\u5b50\u5206\u6bcd\u5404\u9664\u4ee5cos²(x/2 + \u03c0/4)
= \u222b sec²(x/2 + \u03c0/4)/tan(x/2 + \u03c0/4) d(x/2)
= \u222b 1/tan(x/2 + \u03c0/4) d[tan(x/2 + \u03c0/4)]
= ln|tan(x/2 + \u03c0/4)| + C
\u4ed6\u4eec\u7684\u7b54\u6848\u5f62\u5f0f\u53ef\u4ee5\u4e92\u76f8\u8f6c\u5316\u7684.
(\u221a(1-1/(x+1)))'=(1/(x+1)²)/2\u221a(1-1/(x+1))
=1/2(x+1)\u221a(x²+x)
\u5148\u5206\u90e8\u79ef\u5206\u6cd5
=xln(1+\u221a(1-1/(x+1))-\u222bx/2(x+1)\u221a(x²+x)dx
\u7136\u540e\u540e\u9762\u79ef\u5206\u90e8\u5206\u4e09\u89d2\u6362\u5143\u8131\u6839\u53f7\uff0c
\u4ee4x=secu/2-1/2
\u79ef\u5206\u90e8\u5206=\u222b(secu-1)/(secu+1)tanud(secu/2-1/2)
=\u222bsecu(secu-1)²/tan²udu
=\u222bsecucsc²u-2csc²u+cscucotudu
=-\u222bsecudcotu+2cotu-cscu
=2cotu-cscu-secucotu+\u222bcotudsecu
=2cotu-cscu-cscu+\u222bsecudu
=2cotu-2cscu+ln|tanu+secu|+C
\u5c06cotu=1/tanu=1/2\u221a(x²+x)
cscu=(secu/2)/(tanu/2)=(x+1/2)/\u221a(x²+x)\u4ee3\u5165\u6574\u5408
∫10^(arctanx) dx
=x 10^(arctanx) - ∫x · 10^(arctanx) · ln10 · 1/(1+x²) dx
=x 10^(arctanx) - ln10 ∫x · 10^(arctanx)/(1+x²) dx
=x 10^(arctanx) - ½ ln10 ∫10^(arctanx)/(1+x²) d(1+x²)
=x 10^(arctanx) - ½ ln10 ∫10^(arctanx)d(arctanx)
=x 10^(arctanx) - ½ ln10 · 10^(arctanx)/ln10 +C
=x 10^(arctanx) - ½ · 10^(arctanx) +C
如图所示：
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