Questions in category: 定积分 (Definite Integral)
分析 >> 数学分析 >> 定积分
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1. 求定积分 0π2sin5θcos4θdθ.

Posted by haifeng on 2025-04-19 10:14:14 last update 2025-04-19 10:14:14 | Answers (2) | 收藏


求定积分

0π2sin5θcos4θdθ.

2. 计算定积分 011t1+tdt.

Posted by haifeng on 2025-04-12 18:31:43 last update 2025-04-12 18:31:43 | Answers (2) | 收藏


计算定积分 011t1+tdt.

3. F(x) 定义为: F(x)=0xf(t)dtx, 当 x0 时; F(0)=0, 这里 fR 上的连续函数, 且 limx0f(x)x=1. 证明 F(x)x=0 处连续.

Posted by haifeng on 2025-02-28 22:25:34 last update 2025-02-28 22:33:53 | Answers (1) | 收藏


F(x)={0xf(t)dtx,x0,0,x=0. 其中 fC(R), 且 limx0f(x)x=1

证明: F(x)x=0 处连续.

4. f(x) 为连续函数, 且 f(x)=x+02f(x)dx, 求 f(x).

Posted by haifeng on 2024-12-01 21:00:01 last update 2024-12-01 21:02:33 | Answers (1) | 收藏


f(x) 为连续函数, 且 f(x)=x+02f(x)dx, 求 f(x).

5. 计算积分 0πxsinx1+cos2xdx.

Posted by haifeng on 2024-11-25 23:26:59 last update 2024-11-25 23:26:59 | Answers (2) | 收藏


f(x)[0,1] 上的连续函数, 证明

0πxf(sinx)dx=π20πf(sinx)dx .

利用这个等式计算积分 0πxsinx1+cos2xdx.

6. f(x)[a,b] 上可积, g(x)f(x) 只在有限个点处不同, 则 g(x)[a,b] 上也可积, 并且它们的积分相等.

Posted by haifeng on 2024-11-20 10:41:38 last update 2024-11-20 10:41:38 | Answers (1) | 收藏


f(x)[a,b] 上可积, g(x)f(x) 只在有限个点处不同, 则 g(x)[a,b] 上也可积, 且

abf(x)dx=abg(x)dx.

7. Sn=1+12+13++1n, 求 [S2024].

Posted by haifeng on 2024-05-04 21:44:33 last update 2024-05-29 18:30:44 | Answers (2) | 收藏


Sn=1+12+13++1n,

[S2024].

8. 求下列定积分

Posted by haifeng on 2024-04-18 16:37:50 last update 2024-04-18 16:37:50 | Answers (1) | 收藏


1.    011(1+x2)2dx.

9. 求积分 01x4(1x)41+x2dx

Posted by haifeng on 2023-11-22 13:00:17 last update 2023-11-22 13:00:17 | Answers (1) | 收藏


证明:

01x4(1x)41+x2dx=227π

 

 

参考:

知乎大佬是怎么"注意到"这么恐怖的积分的?_哔哩哔哩_bilibili

10. 计算 In=e101xnexdx.

Posted by haifeng on 2023-10-28 19:58:58 last update 2023-10-28 20:37:58 | Answers (1) | 收藏


计算 In=e101xnexdx (n=0,1,2,)

 

容易证明

In={1nIn1,n1,1e1,n=0.

将递推公式改写为 In+1=1(n+1)In, 则使用 Sowya 可以计算如下:

首先计算 I0:

 

>> setprecision(100)
Now the precision is: 100

------------------------

>> exp(1)
out> 2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274

 

令 e 等于上面的 exp(1)

>> e=2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274
 

>> 1-1/e
in> 1-1/2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274

out> 0.6321205588285576784044762298385391325541888689682321654921631983025385042551001966428527256540803562
------------------------

然后再计算 I1=1I0:

>> 1-0.6321205588285576784044762298385391325541888689682321654921631983025385042551001966428527256540803562
in> 1-0.6321205588285576784044762298385391325541888689682321654921631983025385042551001966428527256540803562

out> 0.3678794411714423215955237701614608674458111310317678345078368016974614957448998033571472743459196438
------------------------


利用 printRecursiveSeries() 函数(Pro版本中提供)计算 In, n=1,2,3,,20.

>> printRecursiveSeries(1-(n+1)*I_n,I_n,0.3678794411714423215955237701614608674458111310317678345078368016974614957448998033571472743459196438,20,\n,linenumber)
[1]     0.3678794411714423215955237701614608674458111310317678345078368016974614957448998033571472743459196438
[2]     0.2642411176571153568089524596770782651083777379364643309843263966050770085102003932857054513081607124
[3]     0.2072766470286539295731426209687652046748667861906070070470208101847689744693988201428836460755178628
[4]     0.1708934118853842817074295161249391813005328552375719718119167592609241021224047194284654156979285488
[5]     0.1455329405730785914628524193753040934973357238121401409404162036953794893879764028576729215103572560
[6]     0.1268023565615284512228854837481754390159856571271591543575027778277230636721415828539624709378564640
[7]     0.1123835040693008414398016137627719268881004001098859194974805552059385542950089200222627034350047520
[8]     0.1009319674455932684815870898978245848951967991209126440201555583524915656399286398218983725199619840
[9]     0.0916122929896605836657161909195787359432288079117862038185999748275759092406422416029146473203421440
[10]    0.0838770701033941633428380908042126405677119208821379618140002517242409075935775839708535267965785600
[11]    0.0773522288626642032287810011536609537551688702964824200459972310333500164706465763206112052376358400
[12]    0.0717732536480295612546279861560685549379735564422109594480332275997998023522410841526655371483699200
[13]    0.0669477025756157036898361799711087858063437662512575271755680412026025694208659060153480170711910400
[14]    0.0627321639413801483422934804044769987111872724823946195420474231635640281078773157851277610033254400
[15]    0.0590175408792977748655977939328450193321909127640807068692886525465395783818402632230835849501184000
[16]    0.0557193459312356021504352970744796906849453957747086900913815592553667458905557884306626407981056000
[17]    0.0527711191689947634425999497338452583559282718299522684465134926587653198605515966787351064322048000
[18]    0.0501198549580942580332009047907853495932911070608591679627571321422242425100712597827680842203136000
[19]    0.0477227557962090973691828089750783577274689658436758087076144892977393923086460641274063998140416000
[20]    0.0455448840758180526163438204984328454506206831264838258477102140452121538270787174518720037191680000


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