求sinxsin2xsin3x的不定积分,要具体步骤 ∫sin^3(x) dx 求不定积分
sinxsin2xsin3x\u7684\u4e0d\u5b9a\u79ef\u5206\u222bsin^3(x) dx \u6c42\u4e0d\u5b9a\u79ef\u5206\u4e3a1/3cos³x-cosx+C
\u89e3\uff1a\u222bsin^3(x) dx
=\u222bsin^2(x)*sinxdx
=\u222b\uff081-cos^2(x)\uff09d\uff08-cosx\uff09
=\u222b\uff08cos^2(x)-1\uff09dcosx
=\u222bcos^2(x)dcosx-\u222b1dcosx
=1/3cos^3(x)-cosx+C
\u6269\u5c55\u8d44\u6599
\u6027\u8d28
1\u3001\u51fd\u6570\u7684\u548c\u7684\u4e0d\u5b9a\u79ef\u5206\u7b49\u4e8e\u5404\u4e2a\u51fd\u6570\u7684\u4e0d\u5b9a\u79ef\u5206\u7684\u548c\uff1b\u5373\uff1a\u8bbe\u51fd\u6570
\u53ca
\u7684\u539f\u51fd\u6570\u5b58\u5728\uff0c\u5219
2\u3001\u6c42\u4e0d\u5b9a\u79ef\u5206\u65f6\uff0c\u88ab\u79ef\u51fd\u6570\u4e2d\u7684\u5e38\u6570\u56e0\u5b50\u53ef\u4ee5\u63d0\u5230\u79ef\u5206\u53f7\u5916\u9762\u6765\u3002\u5373\uff1a\u8bbe\u51fd\u6570
\u7684\u539f\u51fd\u6570\u5b58\u5728\uff0c
\u975e\u96f6\u5e38\u6570\uff0c\u5219
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u4e0d\u5b9a\u79ef\u5206
求sinxsin2xsin3x的不定积分的解答过程如下:
运用公式:
sinα·sinβ=-(1/2)[cos(α+β)-cos(α-β)]
sin2α=2sinαcosα
积化和差公式:
sinα·cosβ=(1/2)[sin(α+β)+sin(α-β)]
cosα·sinβ=(1/2)[sin(α+β)-sin(α-β)]
cosα·cosβ=(1/2)[cos(α+β)+cos(α-β)]
扩展资料:
分部积分:
(uv)'=u'v+uv'
得:u'v=(uv)'-uv'
两边积分得:∫ u'v dx=∫ (uv)' dx - ∫ uv' dx
即:∫ u'v dx = uv - ∫ uv' d,这就是分部积分公式
也可简写为:∫ v du = uv - ∫ u dv
常用积分公式:
1)∫0dx=c
2)∫x^udx=(x^(u+1))/(u+1)+c
3)∫1/xdx=ln|x|+c
4)∫a^xdx=(a^x)/lna+c
5)∫e^xdx=e^x+c
6)∫sinxdx=-cosx+c
7)∫cosxdx=sinx+c
8)∫1/(cosx)^2dx=tanx+c
9)∫1/(sinx)^2dx=-cotx+c
10)∫1/√(1-x^2) dx=arcsinx+c
求sinxsin2xsin3x的不定积分的解答过程如下:
运用公式:
sinα·sinβ=-(1/2)[cos(α+β)-cos(α-β)]
sin2α=2sinαcosα
扩展资料:
积化和差公式:
sinα·cosβ=(1/2)[sin(α+β)+sin(α-β)]
cosα·sinβ=(1/2)[sin(α+β)-sin(α-β)]
cosα·cosβ=(1/2)[cos(α+β)+cos(α-β)]
二倍角公式
sin2α=2sinαcosα
tan2α=2tanα/(1-tan^2(α))
cos2α=cos^2(α)-sin^2(α)=2cos^2(α)-1=1-2sin^2(α)
不定积分的公式
1、∫ a dx = ax + C,a和C都是常数
2、∫ x^a dx = [x^(a + 1)]/(a + 1) + C,其中a为常数且 a ≠ -1
3、∫ 1/x dx = ln|x| + C
4、∫ a^x dx = (1/lna)a^x + C,其中a > 0 且 a ≠ 1
5、∫ e^x dx = e^x + C
6、∫ cosx dx = sinx + C
7、∫ sinx dx = - cosx + C
8、∫ cotx dx = ln|sinx| + C = - ln|cscx| + C
9、∫ tanx dx = - ln|cosx| + C = ln|secx| + C
10、∫ secx dx =ln|cot(x/2)| + C = (1/2)ln|(1 + sinx)/(1 - sinx)| + C = - ln|secx - tanx| + C = ln|secx + tanx| + C
利用三角函数的积化和差公式即可
∫sinxsin2xsin3xdx
=∫sinx * (1/2)(cosx-cos5x)dx
=(1/2)∫sinxcosxdx - (1/2)∫sinxcos5xdx
=(1/2)∫sinxdsinx - (1/4)∫(sinx6x-sin4x)dx
=(1/2)(sin�0�5/2) - (1/4)(-1/6)cos6x + (1/4)(-1/4)cos4x + C
= (1/4)sin�0�5x + (1/24)cos6x - (1/16)cos4x + C
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