sinx的三次方的不定积分，用换元积分求 sinx的四次方求不定积分？？ 过程
\u7528\u6362\u5143\u79ef\u5206\u6cd5\u6c42sinx/\uff081+cosx\uff09^3\u7684\u4e0d\u5b9a\u79ef\u5206\u600e\u4e48\u6c42\uff0c\u6c42\u5e2e\u5fd9\u3002\u4ee4cosx=t\uff0c\u5219dt=-sinxdx
\u5219\u539f\u5f0f\u53ef\u5316\u4e3a
\u222b-dt/(1+t)^3
=1/[2(1+t)^2]+c
\u5c06t\u5e26\u6362\u56de\u6765
\u539f\u5f0f=1/[2(1+cosx)^2]+c
\u222b(sinx)^4dx\u7684\u4e0d\u5b9a\u79ef\u5206\u4e3a3/8*x-1/4cosx*(sinx)^3+3/8*sinx*cosx+C\u3002
\u89e3\uff1a\u222b(sinx)^4dx
=\u222b(sinx)^3*sinxdx
=-\u222b(sinx)^3*dcosx
=-cosx*(sinx)^3+\u222bcosxd(sinx)^3
=-cosx*(sinx)^3+3\u222bcosx*cosx*(sinx)^2dx
=-cosx*(sinx)^3+3\u222b(cosx)^2*(sinx)^2dx
=-cosx*(sinx)^3+3\u222b(1-(sinx)^2)*(sinx)^2dx
=-cosx*(sinx)^3+3\u222b(sinx)^2dx-3\u222b(sinx)^4dx
\u5219\uff0c4\u222b(sinx)^4dx=-cosx*(sinx)^3+3\u222b(sinx)^2dx
=-cosx*(sinx)^3+3/2\u222b(1-cos2x)dx
=-cosx*(sinx)^3+3/2*x-3/2\u222bcos2xdx
=-cosx*(sinx)^3+3/2*x-3/4*sin2x+C
=3/2*x-cosx*(sinx)^3+3/2*sinx*cosx+C
\u5f97\uff0c\u222b(sinx)^4dx=3/8*x-1/4cosx*(sinx)^3+3/8*sinx*cosx+C
\u6269\u5c55\u8d44\u6599\uff1a
1\u3001\u5206\u90e8\u79ef\u5206\u6cd5\u662f\u5fae\u79ef\u5206\u5b66\u4e2d\u7684\u4e00\u7c7b\u91cd\u8981\u7684\u3001\u57fa\u672c\u7684\u8ba1\u7b97\u79ef\u5206\u7684\u65b9\u6cd5\u3002
2\u3001\u5206\u90e8\u79ef\u5206\u6cd5\u7684\u516c\u5f0f\u4e3a\uff1a\u222bu\uff08x\uff09v'\uff08x\uff09dx=\u222bu\uff08x\uff09dv\uff08x\uff09=u\uff08x\uff09*v\uff08x\uff09-\u222bv\uff08x\uff09du\uff08x\uff09
3\u3001\u5206\u90e8\u79ef\u5206\u4e2d\u5e38\u89c1\u5f62\u5f0f
\uff081\uff09\u6c42\u542b\u6709e^x\u7684\u51fd\u6570\u7684\u79ef\u5206
\u222bx*e^xdx=\u222bxd(e^x)=x*e^x-\u222be^xdx
\uff082\uff09\u6c42\u542b\u6709\u4e09\u89d2\u51fd\u6570\u7684\u51fd\u6570\u7684\u79ef\u5206
\u222bx*cosxdx=\u222bx*d(sinx)=x*sinx-\u222bsinxdx
\uff083\uff09\u6c42\u542b\u6709arctanx\u7684\u51fd\u6570\u7684\u79ef\u5206
\u222bx*arctanxdx=1/2\u222barctanxd(x^2)=1/2(x^2)*arctanx-1/2\u222b(x^2)d(arctanx)
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u5206\u90e8\u79ef\u5206\u6cd5
= ∫ (1-(cosx)^2) (-1) d(cosx)
= - cosx +1/3 (cosx)^3 + C
还可以有别的计算方法，得到的结果外型上可能会有区别，但都是对的（因为三角函数加上或者减去常数会变成不同的形式）
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