求不定积分1 求不定积分∫(1+x^2)^1/2dx
\u6c421/(1+x^3)\u7684\u4e0d\u5b9a\u79ef\u5206\u8be6\u7ec6\u7684\u89e3\u9898\u8fc7\u7a0b\u5982\u4e0b\uff1a
\u62d3\u5c55\u5185\u5bb9\uff1a
\u5728\u5fae\u79ef\u5206\u4e2d\uff0c\u4e00\u4e2a\u51fd\u6570f \u7684\u4e0d\u5b9a\u79ef\u5206\uff0c\u6216\u539f\u51fd\u6570\uff0c\u6216\u53cd\u5bfc\u6570\uff0c\u662f\u4e00\u4e2a\u5bfc\u6570\u7b49\u4e8ef \u7684\u51fd\u6570 F \uff0c\u5373F \u2032 = f\u3002\u4e0d\u5b9a\u79ef\u5206\u548c\u5b9a\u79ef\u5206\u95f4\u7684\u5173\u7cfb\u7531\u5fae\u79ef\u5206\u57fa\u672c\u5b9a\u7406\u786e\u5b9a\u3002\u5176\u4e2dF\u662ff\u7684\u4e0d\u5b9a\u79ef\u5206\u3002\u8fd9\u6837\uff0c\u8bb8\u591a\u51fd\u6570\u7684\u5b9a\u79ef\u5206\u7684\u8ba1\u7b97\u5c31\u53ef\u4ee5\u7b80\u4fbf\u5730\u901a\u8fc7\u6c42\u4e0d\u5b9a\u79ef\u5206\u6765\u8fdb\u884c\u3002
\u8bbeF(x)\u662f\u51fd\u6570f(x)\u7684\u4e00\u4e2a\u539f\u51fd\u6570\uff0c\u6211\u4eec\u628a\u51fd\u6570f(x)\u7684\u6240\u6709\u539f\u51fd\u6570F(x)+ C(C\u4e3a\u4efb\u610f\u5e38\u6570\uff09\u53eb\u505a\u51fd\u6570f(x)\u7684\u4e0d\u5b9a\u79ef\u5206\uff0c\u8bb0\u4f5c\u222bf(x)dx\u6216\u8005\u222bf\uff08\u9ad8\u7b49\u5fae\u79ef\u5206\u4e2d\u5e38\u7701\u53bbdx\uff09\uff0c\u5373\u222bf(x)dx=F(x)+C\u3002
\u5176\u4e2d\u222b\u53eb\u505a\u79ef\u5206\u53f7,f(x)\u53eb\u505a\u88ab\u79ef\u51fd\u6570,x\u53eb\u505a\u79ef\u5206\u53d8\u91cf,f(x)dx\u53eb\u505a\u88ab\u79ef\u5f0f,C\u53eb\u505a\u79ef\u5206\u5e38\u6570\uff0c\u6c42\u5df2\u77e5\u51fd\u6570\u7684\u4e0d\u5b9a\u79ef\u5206\u7684\u8fc7\u7a0b\u53eb\u505a\u5bf9\u8fd9\u4e2a\u51fd\u6570\u8fdb\u884c\u79ef\u5206\u3002
\u4ee4x=tan(t), \u5219dx=(sect)^2dt
\u5e26\u5165\u222b(1+x^2)^(1/2)dx
=\u222bsectdtant
=secttant-\u222btantdsect
=sect*tant-\u222bsect*tan²tdt
=sect*tant-\u222bsect(sec²t-1)dt
=secttant-\u222bsec³tdt+\u222bsectdt
=secttant-\u222bsec³tdt+ln|sect+tant|
2\u222bsec³tdt=secttant+ln|sect+tant|
\u222bsec³tdt=(secttant+ln|sect+tant|)/2+C
\u53cd\u5e26\u56de\u5f97\uff1a
\u222b(1+x^2)^1/2dx
=(x\u221a(1+x^2)+ln|x+\u221a(1+x^2)|)/2+C
\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\uff0cb]\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
\u6269\u5c55\u8d44\u6599\uff1a
\u6c42\u51fd\u6570f(x)\u7684\u4e0d\u5b9a\u79ef\u5206\uff0c\u5c31\u662f\u8981\u6c42\u51faf(x)\u7684\u6240\u6709\u7684\u539f\u51fd\u6570\uff0c\u7531\u539f\u51fd\u6570\u7684\u6027\u8d28\u53ef\u77e5\uff0c\u53ea\u8981\u6c42\u51fa\u51fd\u6570f(x)\u7684\u4e00\u4e2a\u539f\u51fd\u6570\uff0c\u518d\u52a0\u4e0a\u4efb\u610f\u7684\u5e38\u6570C\u5c31\u5f97\u5230\u51fd\u6570f(x)\u7684\u4e0d\u5b9a\u79ef\u5206\u3002
\u5982\u679cF(x)\u662ff(x)\u5728\u533a\u95f4I\u4e0a\u7684\u4e00\u4e2a\u539f\u51fd\u6570\uff0c\u90a3\u4e48F(x)+C\u5c31\u662ff(x)\u7684\u4e0d\u5b9a\u79ef\u5206\uff0c\u5373\u222bf(x)dx=F(x)+C\u3002\u56e0\u800c\u4e0d\u5b9a\u79ef\u5206\u222bf(x) dx\u53ef\u4ee5\u8868\u793af(x)\u7684\u4efb\u610f\u4e00\u4e2a\u539f\u51fd\u6570\u3002
\u8bbef(x)\u5728\u533a\u95f4[a\uff0cb]\u4e0a\u8fde\u7eed\uff0c\u5219f(x)\u5728[a\uff0cb]\u4e0a\u53ef\u79ef\u3002\u8bbef(x)\u533a\u95f4[a\uff0cb]\u4e0a\u6709\u754c\uff0c\u4e14\u53ea\u6709\u6709\u9650\u4e2a\u95f4\u65ad\u70b9\uff0c\u5219f(x)\u5728[a\uff0cb]\u4e0a\u53ef\u79ef\u3002\u8bbef(x)\u5728\u533a\u95f4[a\uff0cb]\u4e0a\u5355\u8c03\uff0c\u5219f(x)\u5728[a\uff0cb]\u4e0a\u53ef\u79ef\u3002
\u4e00\u4e2a\u5b9a\u79ef\u5206\u5f0f\u7684\u503c\uff0c\u5c31\u662f\u539f\u51fd\u6570\u5728\u4e0a\u9650\u7684\u503c\u4e0e\u539f\u51fd\u6570\u5728\u4e0b\u9650\u7684\u503c\u7684\u5dee\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1\u2014\u2014\u4e0d\u5b9a\u79ef\u5206
实际上，写到倒数第二步就可以了。
以上，请采纳。
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