求不定积分∫larccosxdx=？ ∫arccosxdx=?
\u222barccosxdx=? \u7528\u5206\u90e8\u79ef\u5206\u6cd5\u6c42\u222barccosxdx=xarccosx-\u221a(1-x²)+C\u3002C\u4e3a\u79ef\u5206\u5e38\u6570\u3002
\u89e3\u7b54\u8fc7\u7a0b\u5982\u4e0b\uff1a
\u222barccosxdx
=xarccosx-\u222bxdarccosx
=xarccosx+\u222bxdx/\u221a(1-x²)
=xarccosx-\u222bd(1-x²)/2\u221a(1-x²)
=xarccosx-\u221a(1-x²)+C
\u6269\u5c55\u8d44\u6599\uff1a
\u5206\u90e8\u79ef\u5206\uff1a
(uv)'=u'v+uv'
\u5f97\uff1au'v=(uv)'-uv'
\u4e24\u8fb9\u79ef\u5206\u5f97\uff1a\u222b u'v dx=\u222b (uv)' dx - \u222b uv' dx
\u5373\uff1a\u222b u'v dx = uv - \u222b uv' d,\u8fd9\u5c31\u662f\u5206\u90e8\u79ef\u5206\u516c\u5f0f
\u4e5f\u53ef\u7b80\u5199\u4e3a\uff1a\u222b v du = uv - \u222b u dv
\u5e38\u7528\u79ef\u5206\u516c\u5f0f\uff1a
1\uff09\u222b0dx=c
2\uff09\u222bx^udx=(x^(u+1))/(u+1)+c
3\uff09\u222b1/xdx=ln|x|+c
4\uff09\u222ba^xdx=(a^x)/lna+c
5\uff09\u222be^xdx=e^x+c
6\uff09\u222bsinxdx=-cosx+c
7\uff09\u222bcosxdx=sinx+c
8\uff09\u222b1/(cosx)^2dx=tanx+c
9\uff09\u222b1/(sinx)^2dx=-cotx+c
10\uff09\u222b1/\u221a\uff081-x^2) dx=arcsinx+c
\u89e3\uff1a\u222barccosxdx
=xarccosx-\u222bxd(arccosx)
=xarccosx+\u222bxdx/\u221a(1-x²)
=xarccosx+(1/2)\u222bd(1-x²)/\u221a(1-x²)
=xarccosx+(1/2)\u222b[(1-x²)^(-1/2)]d(1-x²)
=xarccosx+\u221a(1-x²)+C \u671b\u91c7\u7eb3\uff01
看图：
∫arccosxdx
=xarccosx+∫x/根号（1-x^2)dx
=xarccosx-1/2∫x/根号（1-x^2)d1-x^2
=xarccosx-根号（1-x^2)+c
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