求数学大神帮忙！！求x·arccosx的不定积分！！非常感谢啦 求xarccosx的不定积分
\u6c42arccosx\u7684\u4e0d\u5b9a\u79ef\u5206\u53ef\u4ee5\u7528\u53cd\u51fd\u6570\u6765\u505a
y=arccosx,
\u222barccosxdx=\u222bydcosy=ycosy-\u222bcosydy
=ycosy-siny+C
=xarccosx-\u221a(1-x^2)+C
\u6269\u5c55\u8d44\u6599\u4e0d\u5b9a\u79ef\u5206\u7684\u516c\u5f0f
1\u3001\u222b a dx = ax + C\uff0ca\u548cC\u90fd\u662f\u5e38\u6570
2\u3001\u222b x^a dx = [x^(a + 1)]/(a + 1) + C\uff0c\u5176\u4e2da\u4e3a\u5e38\u6570\u4e14 a \u2260 -1
3\u3001\u222b 1/x dx = ln|x| + C
4\u3001\u222b a^x dx = (1/lna)a^x + C\uff0c\u5176\u4e2da > 0 \u4e14 a \u2260 1
5\u3001\u222b e^x dx = e^x + C
6\u3001\u222b cosx dx = sinx + C
7\u3001\u222b sinx dx = - cosx + C
8\u3001\u222b cotx dx = ln|sinx| + C = - ln|cscx| + C
\u5148\u7528\u5206\u90e8\u79ef\u5206
\u222bx\u00b7arccosx dx
=x²/2\u00b7arccosx-\u222bx²/2\u00b7[-1/\u221a(1-x²)] dx
=x²/2\u00b7arccosx+1/2 \u222bx²/\u221a(1-x²) dx
\u4e0b\u9762\u6c42 \u222bx²/\u221a(1-x²) dx
\u4ee4sint=x\uff0c\u5219dx=cost dt
\u222bx²/\u221a(1-x²) dx
=\u222bsin²t/cost \u00b7costdt
=\u222bsin²t dt
=\u222b(1-cos2t)/2 dt
=t-1/4\u00b7sin2t+C
=arcsinx-1/2\u00b7x\u221a(1-x²)+C
\u2234 \u222bx\u00b7arccosx dx=x²/2\u00b7arccosx+1/2\u00b7arcsinx-1/4\u00b7x\u221a(1-x²)+C
\u6269\u5c55\u8d44\u6599\uff1a
\u5b9a\u79ef\u5206\u662f\u4e00\u4e2a\u6570\uff0c\u800c\u4e0d\u5b9a\u79ef\u5206\u662f\u4e00\u4e2a\u8868\u8fbe\u5f0f\uff0c\u5b83\u4eec\u4ec5\u4ec5\u662f\u6570\u5b66\u4e0a\u6709\u4e00\u4e2a\u8ba1\u7b97\u5173\u7cfb\u3002\u4e00\u4e2a\u51fd\u6570\uff0c\u53ef\u4ee5\u5b58\u5728\u4e0d\u5b9a\u79ef\u5206\uff0c\u800c\u4e0d\u5b58\u5728\u5b9a\u79ef\u5206\uff0c\u4e5f\u53ef\u4ee5\u5b58\u5728\u5b9a\u79ef\u5206\uff0c\u800c\u6ca1\u6709\u4e0d\u5b9a\u79ef\u5206\u3002
\u8fde\u7eed\u51fd\u6570\uff0c\u4e00\u5b9a\u5b58\u5728\u5b9a\u79ef\u5206\u548c\u4e0d\u5b9a\u79ef\u5206\uff1b\u82e5\u5728\u6709\u9650\u533a\u95f4[a,b]\u4e0a\u53ea\u6709\u6709\u9650\u4e2a\u95f4\u65ad\u70b9\u4e14\u51fd\u6570\u6709\u754c\uff0c\u5219\u5b9a\u79ef\u5206\u5b58\u5728\uff1b\u82e5\u6709\u8df3\u8dc3\u3001\u53ef\u53bb\u3001\u65e0\u7a77\u95f4\u65ad\u70b9\uff0c\u5219\u539f\u51fd\u6570\u4e00\u5b9a\u4e0d\u5b58\u5728\uff0c\u5373\u4e0d\u5b9a\u79ef\u5206\u4e00\u5b9a\u4e0d\u5b58\u5728\u3002
法一：
先用分部积分
∫x·arccosx dx
=x²/2·arccosx-∫x²/2·[-1/√(1-x²)] dx
=x²/2·arccosx+1/2 ∫x²/√(1-x²) dx
下面求 ∫x²/√(1-x²) dx
令sint=x，则dx=cost dt
∫x²/√(1-x²) dx
=∫sin²t/cost ·costdt
=∫sin²t dt
=∫(1-cos2t)/2 dt
=t-1/4·sin2t+C
=arcsinx-1/2·x√(1-x²)+C
∴ ∫x·arccosx dx=x²/2·arccosx+1/2·arcsinx-1/4·x√(1-x²)+C
法二：
令arccosx=t，则x=cost，dx=-sint dt
∫x·arccosx dx
=∫cost·t·(-sint)dt
=-1/2∫sin2t·t dt
=-1/2[(-cos2t)/2·t+1/2∫cos2tdt]
=-1/2[(-cos2t)/2·t+1/4·sin2t]+C
=1/4·cos2t·t-1/8·sint2t+C
=x²/2·arccosx+1/2·arcsinx-1/4·x√(1-x²)+C
令arccosx=t，则cost=x，待进去就好了
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