Интеграл sin(x)^(42) (dx)

Вы ввели

  1            
  /            
 |             
 |     42      
 |  sin  (x) dx
 |             
/              
0              

$$\int_{0}^{1} \sin^{42}{\left (x \right )}\, dx$$

Ответ

  1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
  /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
 |                                                                          3                            7                           5                           9                           11                          27                          13                          21                         19                         23                         25                         15                         17                        29                       31                      33                      35                    37                   39                41          
 |     42         67282234305    67282234305*cos(1)*sin(1)   22427411435*sin (1)*cos(1)   13456446861*sin (1)*cos(1)   4485482287*sin (1)*cos(1)   4485482287*sin (1)*cos(1)   2038855585*sin  (1)*cos(1)   195020969*sin  (1)*cos(1)   156835045*sin  (1)*cos(1)   156835045*sin  (1)*cos(1)   94101027*sin  (1)*cos(1)   75008065*sin  (1)*cos(1)   45004839*sin  (1)*cos(1)   31367009*sin  (1)*cos(1)   31367009*sin  (1)*cos(1)   6724861*sin  (1)*cos(1)   216931*sin  (1)*cos(1)   19721*sin  (1)*cos(1)   19721*sin  (1)*cos(1)   533*sin  (1)*cos(1)   41*sin  (1)*cos(1)   sin  (1)*cos(1)
 |  sin  (x) dx = ------------ - ------------------------- - -------------------------- - -------------------------- - ------------------------- - ------------------------- - -------------------------- - ------------------------- - ------------------------- - ------------------------- - ------------------------ - ------------------------ - ------------------------ - ------------------------ - ------------------------ - ----------------------- - ---------------------- - --------------------- - --------------------- - ------------------- - ------------------ - ---------------
 |                549755813888          549755813888                274877906944                 240518168576                 68719476736                 90194313216                 45097156608                   6667960320                  3758096384                  4741660672                 2709520384                 2370830336                 1481768960                 805306368                  855638016                  238141440                 7938048                   744192                  766080                 21280                 1680                 42      
/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   

$${{21637803915\,\sin 32+1298201657041800\,\sin 16+13674528\,\sin ^58 -60489700037760\,\sin ^38+25955945044968960\,\sin 8+66448076800\, \sin ^94-50342147850240\,\sin ^74+2875313840652288\,\sin ^54- 30299958505635840\,\sin ^34+111869703880704000\,\sin 4-20321402880\, \sin ^{21}2+4941309542400\,\sin ^{19}2-198815042764800\,\sin ^{17}2+ 3124488849850368\,\sin ^{15}2-25236256094945280\,\sin ^{13}2+ 119298665176104960\,\sin ^{11}2-353477526447718400\,\sin ^92+ 679209564163276800\,\sin ^72-855804050845728768\,\sin ^52+ 708508800264437760\,\sin ^32-447479242272276480\,\sin 2+ 219060189739591200}\over{1789916969089105920}}$$

Численный ответ

2.45252299698723e-5

Ответ (Неопределённый)

  /                                                                                                                                                                                                                                                                                                                                                                                                          
 |                               3                  3                 7               17               9               13               5                      21              3               5              19              7               15               11                9                                                         5                                                                 
 |    42             13934123*sin (8*x)   284007*sin (4*x)   52849*sin (4*x)   495*sin  (2*x)   455*sin (2*x)   231*sin  (2*x)   153*sin (2*x)   sin(2*x)   sin  (2*x)   19*sin (2*x)    21*sin (8*x)   55*sin  (2*x)   85*sin (2*x)   143*sin  (2*x)   273*sin  (2*x)   1435*sin (4*x)   212667*sin(32*x)   1048575*sin(4*x)   2156067*sin (4*x)   498257459*sin(8*x)   1594920705*sin(16*x)   67282234305*x
 | sin  (x) dx = C - ------------------ - ---------------- - --------------- - -------------- - ------------- - -------------- - ------------- - -------- - ---------- + ------------ + ------------- + ------------- + ------------ + -------------- + -------------- + -------------- + ---------------- + ---------------- + ----------------- + ------------------ + -------------------- + -------------
 |                      412316860416          16777216          1879048192        4456448            2304           16384             320           4        88080384         48        2748779069440      19922944         224            81920             4096         38654705664      17592186044416        16777216           1342177280         34359738368          2199023255552        549755813888
/                                                                                                                                                                                                                                                                                                                                                                                                            

$${{{{21\,\left({{{{5\,\left({{{{\sin \left(32\,x\right)}\over{2}}+16 \,x}\over{8}}+{{\sin \left(16\,x\right)}\over{2}}+4\,x\right)}\over{ 64}}+{{5\,\left({{\sin \left(16\,x\right)}\over{2}}+8\,x\right) }\over{32}}+{{{{\sin ^5\left(8\,x\right)}\over{5}}-{{2\,\sin ^3 \left(8\,x\right)}\over{3}}+\sin \left(8\,x\right)}\over{32}}+{{5\, \left(\sin \left(8\,x\right)-{{\sin ^3\left(8\,x\right)}\over{3}} \right)}\over{16}}+{{5\,\sin \left(8\,x\right)}\over{32}}+{{x}\over{ 4}}}\over{2048}}+{{45\,\left({{{{{{\sin \left(32\,x\right)}\over{2}} +16\,x}\over{8}}+{{\sin \left(16\,x\right)}\over{2}}+4\,x}\over{32}} +{{3\,\left({{\sin \left(16\,x\right)}\over{2}}+8\,x\right)}\over{16 }}+{{\sin \left(8\,x\right)-{{\sin ^3\left(8\,x\right)}\over{3}} }\over{4}}+{{\sin \left(8\,x\right)}\over{4}}+{{x}\over{2}}\right) }\over{2048}}+{{105\,\left({{3\,\left({{\sin \left(16\,x\right) }\over{2}}+8\,x\right)}\over{16}}+{{\sin \left(8\,x\right)-{{\sin ^3 \left(8\,x\right)}\over{3}}}\over{8}}+{{3\,\sin \left(8\,x\right) }\over{8}}+x\right)}\over{1024}}+{{105\,\left({{{{\sin \left(16\,x \right)}\over{2}}+8\,x}\over{8}}+{{\sin \left(8\,x\right)}\over{2}}+ 2\,x\right)}\over{1024}}+{{45\,\left({{\sin \left(8\,x\right)}\over{ 2}}+4\,x\right)}\over{2048}}+{{5\,\left({{\sin ^9\left(4\,x\right) }\over{9}}-{{4\,\sin ^7\left(4\,x\right)}\over{7}}+{{6\,\sin ^5 \left(4\,x\right)}\over{5}}-{{4\,\sin ^3\left(4\,x\right)}\over{3}}+ \sin \left(4\,x\right)\right)}\over{512}}+{{15\,\left(-{{\sin ^7 \left(4\,x\right)}\over{7}}+{{3\,\sin ^5\left(4\,x\right)}\over{5}}- \sin ^3\left(4\,x\right)+\sin \left(4\,x\right)\right)}\over{128}}+ {{63\,\left({{\sin ^5\left(4\,x\right)}\over{5}}-{{2\,\sin ^3\left(4 \,x\right)}\over{3}}+\sin \left(4\,x\right)\right)}\over{256}}+{{15 \,\left(\sin \left(4\,x\right)-{{\sin ^3\left(4\,x\right)}\over{3}} \right)}\over{128}}+{{5\,\sin \left(4\,x\right)}\over{512}}+{{x }\over{256}}\right)}\over{4194304}}+{{665\,\left({{9\,\left({{{{{{ \sin \left(32\,x\right)}\over{2}}+16\,x}\over{8}}+{{\sin \left(16\,x \right)}\over{2}}+4\,x}\over{32}}+{{3\,\left({{\sin \left(16\,x \right)}\over{2}}+8\,x\right)}\over{16}}+{{\sin \left(8\,x\right)-{{ \sin ^3\left(8\,x\right)}\over{3}}}\over{4}}+{{\sin \left(8\,x \right)}\over{4}}+{{x}\over{2}}\right)}\over{1024}}+{{21\,\left({{3 \,\left({{\sin \left(16\,x\right)}\over{2}}+8\,x\right)}\over{16}}+ {{\sin \left(8\,x\right)-{{\sin ^3\left(8\,x\right)}\over{3}}}\over{ 8}}+{{3\,\sin \left(8\,x\right)}\over{8}}+x\right)}\over{256}}+{{63 \,\left({{{{\sin \left(16\,x\right)}\over{2}}+8\,x}\over{8}}+{{\sin \left(8\,x\right)}\over{2}}+2\,x\right)}\over{512}}+{{9\,\left({{ \sin \left(8\,x\right)}\over{2}}+4\,x\right)}\over{256}}+{{{{\sin ^9 \left(4\,x\right)}\over{9}}-{{4\,\sin ^7\left(4\,x\right)}\over{7}}+ {{6\,\sin ^5\left(4\,x\right)}\over{5}}-{{4\,\sin ^3\left(4\,x \right)}\over{3}}+\sin \left(4\,x\right)}\over{512}}+{{9\,\left(-{{ \sin ^7\left(4\,x\right)}\over{7}}+{{3\,\sin ^5\left(4\,x\right) }\over{5}}-\sin ^3\left(4\,x\right)+\sin \left(4\,x\right)\right) }\over{128}}+{{63\,\left({{\sin ^5\left(4\,x\right)}\over{5}}-{{2\, \sin ^3\left(4\,x\right)}\over{3}}+\sin \left(4\,x\right)\right) }\over{256}}+{{21\,\left(\sin \left(4\,x\right)-{{\sin ^3\left(4\,x \right)}\over{3}}\right)}\over{128}}+{{9\,\sin \left(4\,x\right) }\over{512}}+{{x}\over{128}}\right)}\over{2097152}}+{{20349\,\left( {{{{{{{{\sin \left(32\,x\right)}\over{2}}+16\,x}\over{8}}+{{\sin \left(16\,x\right)}\over{2}}+4\,x}\over{32}}+{{3\,\left({{\sin \left(16\,x\right)}\over{2}}+8\,x\right)}\over{16}}+{{\sin \left(8\, x\right)-{{\sin ^3\left(8\,x\right)}\over{3}}}\over{4}}+{{\sin \left(8\,x\right)}\over{4}}+{{x}\over{2}}}\over{512}}+{{7\,\left({{3 \,\left({{\sin \left(16\,x\right)}\over{2}}+8\,x\right)}\over{16}}+ {{\sin \left(8\,x\right)-{{\sin ^3\left(8\,x\right)}\over{3}}}\over{ 8}}+{{3\,\sin \left(8\,x\right)}\over{8}}+x\right)}\over{128}}+{{35 \,\left({{{{\sin \left(16\,x\right)}\over{2}}+8\,x}\over{8}}+{{\sin \left(8\,x\right)}\over{2}}+2\,x\right)}\over{256}}+{{7\,\left({{ \sin \left(8\,x\right)}\over{2}}+4\,x\right)}\over{128}}+{{-{{\sin ^ 7\left(4\,x\right)}\over{7}}+{{3\,\sin ^5\left(4\,x\right)}\over{5}} -\sin ^3\left(4\,x\right)+\sin \left(4\,x\right)}\over{32}}+{{7\, \left({{\sin ^5\left(4\,x\right)}\over{5}}-{{2\,\sin ^3\left(4\,x \right)}\over{3}}+\sin \left(4\,x\right)\right)}\over{32}}+{{7\, \left(\sin \left(4\,x\right)-{{\sin ^3\left(4\,x\right)}\over{3}} \right)}\over{32}}+{{\sin \left(4\,x\right)}\over{32}}+{{x}\over{64 }}\right)}\over{4194304}}+{{14535\,\left({{7\,\left({{3\,\left({{ \sin \left(16\,x\right)}\over{2}}+8\,x\right)}\over{16}}+{{\sin \left(8\,x\right)-{{\sin ^3\left(8\,x\right)}\over{3}}}\over{8}}+{{3 \,\sin \left(8\,x\right)}\over{8}}+x\right)}\over{256}}+{{35\,\left( {{{{\sin \left(16\,x\right)}\over{2}}+8\,x}\over{8}}+{{\sin \left(8 \,x\right)}\over{2}}+2\,x\right)}\over{256}}+{{21\,\left({{\sin \left(8\,x\right)}\over{2}}+4\,x\right)}\over{256}}+{{-{{\sin ^7 \left(4\,x\right)}\over{7}}+{{3\,\sin ^5\left(4\,x\right)}\over{5}}- \sin ^3\left(4\,x\right)+\sin \left(4\,x\right)}\over{128}}+{{21\, \left({{\sin ^5\left(4\,x\right)}\over{5}}-{{2\,\sin ^3\left(4\,x \right)}\over{3}}+\sin \left(4\,x\right)\right)}\over{128}}+{{35\, \left(\sin \left(4\,x\right)-{{\sin ^3\left(4\,x\right)}\over{3}} \right)}\over{128}}+{{7\,\sin \left(4\,x\right)}\over{128}}+{{x }\over{32}}\right)}\over{524288}}+{{146965\,\left({{{{3\,\left({{ \sin \left(16\,x\right)}\over{2}}+8\,x\right)}\over{16}}+{{\sin \left(8\,x\right)-{{\sin ^3\left(8\,x\right)}\over{3}}}\over{8}}+{{3 \,\sin \left(8\,x\right)}\over{8}}+x}\over{128}}+{{15\,\left({{{{ \sin \left(16\,x\right)}\over{2}}+8\,x}\over{8}}+{{\sin \left(8\,x \right)}\over{2}}+2\,x\right)}\over{128}}+{{15\,\left({{\sin \left(8 \,x\right)}\over{2}}+4\,x\right)}\over{128}}+{{3\,\left({{\sin ^5 \left(4\,x\right)}\over{5}}-{{2\,\sin ^3\left(4\,x\right)}\over{3}}+ \sin \left(4\,x\right)\right)}\over{32}}+{{5\,\left(\sin \left(4\,x \right)-{{\sin ^3\left(4\,x\right)}\over{3}}\right)}\over{16}}+{{3\, \sin \left(4\,x\right)}\over{32}}+{{x}\over{16}}\right)}\over{ 2097152}}+{{88179\,\left({{5\,\left({{{{\sin \left(16\,x\right) }\over{2}}+8\,x}\over{8}}+{{\sin \left(8\,x\right)}\over{2}}+2\,x \right)}\over{64}}+{{5\,\left({{\sin \left(8\,x\right)}\over{2}}+4\, x\right)}\over{32}}+{{{{\sin ^5\left(4\,x\right)}\over{5}}-{{2\, \sin ^3\left(4\,x\right)}\over{3}}+\sin \left(4\,x\right)}\over{32}} +{{5\,\left(\sin \left(4\,x\right)-{{\sin ^3\left(4\,x\right)}\over{ 3}}\right)}\over{16}}+{{5\,\sin \left(4\,x\right)}\over{32}}+{{x }\over{8}}\right)}\over{1048576}}+{{101745\,\left({{{{{{\sin \left( 16\,x\right)}\over{2}}+8\,x}\over{8}}+{{\sin \left(8\,x\right) }\over{2}}+2\,x}\over{32}}+{{3\,\left({{\sin \left(8\,x\right) }\over{2}}+4\,x\right)}\over{16}}+{{\sin \left(4\,x\right)-{{\sin ^3 \left(4\,x\right)}\over{3}}}\over{4}}+{{\sin \left(4\,x\right) }\over{4}}+{{x}\over{4}}\right)}\over{2097152}}+{{6783\,\left({{3\, \left({{\sin \left(8\,x\right)}\over{2}}+4\,x\right)}\over{16}}+{{ \sin \left(4\,x\right)-{{\sin ^3\left(4\,x\right)}\over{3}}}\over{8 }}+{{3\,\sin \left(4\,x\right)}\over{8}}+{{x}\over{2}}\right)}\over{ 524288}}+{{5985\,\left({{{{\sin \left(8\,x\right)}\over{2}}+4\,x }\over{8}}+{{\sin \left(4\,x\right)}\over{2}}+x\right)}\over{4194304 }}+{{105\,\left({{\sin \left(4\,x\right)}\over{2}}+2\,x\right) }\over{2097152}}-{{46189\,\sin ^{21}\left(2\,x\right)-510510\,\sin ^{19}\left(2\,x\right)+2567565\,\sin ^{17}\left(2\,x\right)-7759752 \,\sin ^{15}\left(2\,x\right)[phone]\,\sin ^{13}\left(2\,x\right)- 22221108\,\sin ^{11}\left(2\,x\right)[phone]\,\sin ^9\left(2\,x \right)-16628040\,\sin ^7\left(2\,x\right)+8729721\,\sin ^5\left(2\, x\right)-3233230\,\sin ^3\left(2\,x\right)+969969\,\sin \left(2\,x \right)}\over{2034172428288}}+{{21\,\left(12155\,\sin ^{19}\left(2\, x\right)-122265\,\sin ^{17}\left(2\,x\right)+554268\,\sin ^{15} \left(2\,x\right)-1492260\,\sin ^{13}\left(2\,x\right)+2645370\, \sin ^{11}\left(2\,x\right)-3233230\,\sin ^9\left(2\,x\right)+ 2771340\,\sin ^7\left(2\,x\right)-1662804\,\sin ^5\left(2\,x\right)+ 692835\,\sin ^3\left(2\,x\right)-230945\,\sin \left(2\,x\right) \right)}\over{48432676864}}-{{133\,\left(6435\,\sin ^{17}\left(2\,x \right)-58344\,\sin ^{15}\left(2\,x\right)+235620\,\sin ^{13}\left(2 \,x\right)-556920\,\sin ^{11}\left(2\,x\right)+850850\,\sin ^9\left( 2\,x\right)-875160\,\sin ^7\left(2\,x\right)+612612\,\sin ^5\left(2 \,x\right)-291720\,\sin ^3\left(2\,x\right)+109395\,\sin \left(2\,x \right)\right)}\over{5098176512}}+{{2261\,\left(429\,\sin ^{15} \left(2\,x\right)-3465\,\sin ^{13}\left(2\,x\right)+12285\,\sin ^{11 }\left(2\,x\right)-25025\,\sin ^9\left(2\,x\right)+32175\,\sin ^7 \left(2\,x\right)-27027\,\sin ^5\left(2\,x\right)+15015\,\sin ^3 \left(2\,x\right)-6435\,\sin \left(2\,x\right)\right)}\over{ 562298880}}-{{4845\,\left(231\,\sin ^{13}\left(2\,x\right)-1638\, \sin ^{11}\left(2\,x\right)+5005\,\sin ^9\left(2\,x\right)-8580\, \sin ^7\left(2\,x\right)+9009\,\sin ^5\left(2\,x\right)-6006\,\sin ^ 3\left(2\,x\right)+3003\,\sin \left(2\,x\right)\right)}\over{ 149946368}}-{{88179\,\left(-{{\sin ^{11}\left(2\,x\right)}\over{11}} +{{5\,\sin ^9\left(2\,x\right)}\over{9}}-{{10\,\sin ^7\left(2\,x \right)}\over{7}}+2\,\sin ^5\left(2\,x\right)-{{5\,\sin ^3\left(2\,x \right)}\over{3}}+\sin \left(2\,x\right)\right)}\over{524288}}-{{ 146965\,\left({{\sin ^9\left(2\,x\right)}\over{9}}-{{4\,\sin ^7 \left(2\,x\right)}\over{7}}+{{6\,\sin ^5\left(2\,x\right)}\over{5}}- {{4\,\sin ^3\left(2\,x\right)}\over{3}}+\sin \left(2\,x\right) \right)}\over{1048576}}-{{14535\,\left(-{{\sin ^7\left(2\,x\right) }\over{7}}+{{3\,\sin ^5\left(2\,x\right)}\over{5}}-\sin ^3\left(2\,x \right)+\sin \left(2\,x\right)\right)}\over{262144}}-{{20349\,\left( {{\sin ^5\left(2\,x\right)}\over{5}}-{{2\,\sin ^3\left(2\,x\right) }\over{3}}+\sin \left(2\,x\right)\right)}\over{2097152}}-{{665\, \left(\sin \left(2\,x\right)-{{\sin ^3\left(2\,x\right)}\over{3}} \right)}\over{1048576}}-{{21\,\sin \left(2\,x\right)}\over{2097152}} +{{x}\over{1048576}}}\over{2}}$$