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1/(cos²3x)
Solving integrals/ 1/(cos²3x)

Integral of 1/(cos²3x) dx

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The solution

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∫011⋅1cos⁡23(x) dx\int\limits_{0}^{1} 1 \cdot \frac{1}{\cos^{23}{\left(x \right)}}\, dx 
Integral(1/cos(x)^23, (x, 0, 1))
The answer (Indefinite) [src] 
  /                                                                                                                                                                                                                                                                                                                              
 |                                                                                                 9                    13                    5                   17                                    21                   19                   3                    15                    7                    11             
 |      1              88179*log(-1 + sin(x))   88179*log(1 + sin(x))              - 5174056250*sin (x) - 4139920070*sin  (x) - 1551313995*sin (x) - 749786037*sin  (x) - 71957985*sin(x) - 14549535*sin  (x) + 155195040*sin  (x) + 450357600*sin (x) + 2163862272*sin  (x) + 3424523520*sin (x) + 5503713280*sin  (x)          
 | 1*-------- dx = C - ---------------------- + --------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
 |      23                    1048576                  1048576                                     12                     8                     16                    4                   20                  22                   2                    18                     6                     14                     10   
 |   cos  (x)                                                           -86507520 - 39966474240*sin  (x) - 28547481600*sin (x) - 14273740800*sin  (x) - 4757913600*sin (x) - 951582720*sin  (x) + 86507520*sin  (x) + 951582720*sin (x) + 4757913600*sin  (x) + 14273740800*sin (x) + 28547481600*sin  (x) + 39966474240*sin  (x)
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88179 log⁡(sin⁡x+1)1048576−88179 log⁡(sin⁡x−1)1048576−14549535 sin⁡21x−155195040 sin⁡19x+749786037 sin⁡17x−2163862272 sin⁡15x+4139920070 sin⁡13x−5503713280 sin⁡11x+5174056250 sin⁡9x−3424523520 sin⁡7x+1551313995 sin⁡5x−450357600 sin⁡3x+71957985 sin⁡x86507520 sin⁡22x−951582720 sin⁡20x+4757913600 sin⁡18x−14273740800 sin⁡16x+28547481600 sin⁡14x−39966474240 sin⁡12x+39966474240 sin⁡10x−28547481600 sin⁡8x+14273740800 sin⁡6x−4757913600 sin⁡4x+951582720 sin⁡2x−86507520{{88179\,\log \left(\sin x+1\right)}\over{1048576}}-{{88179\,\log \left(\sin x-1\right)}\over{1048576}}-{{14549535\,\sin ^{21}x- 155195040\,\sin ^{19}x+749786037\,\sin ^{17}x-2163862272\,\sin ^{15} x+4139920070\,\sin ^{13}x-5503713280\,\sin ^{11}x+5174056250\,\sin ^ 9x-3424523520\,\sin ^7x+1551313995\,\sin ^5x-450357600\,\sin ^3x+ 71957985\,\sin x}\over{86507520\,\sin ^{22}x-951582720\,\sin ^{20}x+ 4757913600\,\sin ^{18}x-14273740800\,\sin ^{16}x+28547481600\,\sin ^{14}x-39966474240\,\sin ^{12}x+39966474240\,\sin ^{10}x-28547481600 \,\sin ^8x+14273740800\,\sin ^6x-4757913600\,\sin ^4x+951582720\, \sin ^2x-86507520}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002000000
Plot the graph f(x)
The answer [src] 
                                                                             9                    13                    5                   17                                    21                   19                   3                    15                    7                    11             
  88179*log(1 - sin(1))   88179*log(1 + sin(1))              - 5174056250*sin (1) - 4139920070*sin  (1) - 1551313995*sin (1) - 749786037*sin  (1) - 71957985*sin(1) - 14549535*sin  (1) + 155195040*sin  (1) + 450357600*sin (1) + 2163862272*sin  (1) + 3424523520*sin (1) + 5503713280*sin  (1)          
- --------------------- + --------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
         1048576                 1048576                                     12                     8                     16                    4                   20                  22                   2                    18                     6                     14                     10   
                                                  -86507520 - 39966474240*sin  (1) - 28547481600*sin (1) - 14273740800*sin  (1) - 4757913600*sin (1) - 951582720*sin  (1) + 86507520*sin  (1) + 951582720*sin (1) + 4757913600*sin  (1) + 14273740800*sin (1) + 28547481600*sin  (1) + 39966474240*sin  (1)
88179 log⁡(sin⁡1+1)1048576−88179 log⁡(1−sin⁡1)1048576−22720789 sin⁡1712621440 sin⁡221−28835840 sin⁡201+144179200 sin⁡181−432537600 sin⁡161+865075200 sin⁡141−1211105280 sin⁡121+1211105280 sin⁡101−865075200 sin⁡81+432537600 sin⁡61−144179200 sin⁡41+28835840 sin⁡21−2621440−37635637 sin⁡131786432 sin⁡221−8650752 sin⁡201+43253760 sin⁡181−129761280 sin⁡161+259522560 sin⁡141−363331584 sin⁡121+363331584 sin⁡101−259522560 sin⁡81+129761280 sin⁡61−43253760 sin⁡41+8650752 sin⁡21−786432−47036875 sin⁡91786432 sin⁡221−8650752 sin⁡201+43253760 sin⁡181−129761280 sin⁡161+259522560 sin⁡141−363331584 sin⁡121+363331584 sin⁡101−259522560 sin⁡81+129761280 sin⁡61−43253760 sin⁡41+8650752 sin⁡21−786432−88179 sin⁡211524288 sin⁡221−5767168 sin⁡201+28835840 sin⁡181−86507520 sin⁡161+173015040 sin⁡141−242221056 sin⁡121+242221056 sin⁡101−173015040 sin⁡81+86507520 sin⁡61−28835840 sin⁡41+5767168 sin⁡21−524288−9401903 sin⁡51524288 sin⁡221−5767168 sin⁡201+28835840 sin⁡181−86507520 sin⁡161+173015040 sin⁡141−242221056 sin⁡121+242221056 sin⁡101−173015040 sin⁡81+86507520 sin⁡61−28835840 sin⁡41+5767168 sin⁡21−524288−436109 sin⁡1524288 sin⁡221−5767168 sin⁡201+28835840 sin⁡181−86507520 sin⁡161+173015040 sin⁡141−242221056 sin⁡121+242221056 sin⁡101−173015040 sin⁡81+86507520 sin⁡61−28835840 sin⁡41+5767168 sin⁡21−524288+29393 sin⁡19116384 sin⁡221−180224 sin⁡201+901120 sin⁡181−2703360 sin⁡161+5406720 sin⁡141−7569408 sin⁡121+7569408 sin⁡101−5406720 sin⁡81+2703360 sin⁡61−901120 sin⁡41+180224 sin⁡21−16384+85295 sin⁡3116384 sin⁡221−180224 sin⁡201+901120 sin⁡181−2703360 sin⁡161+5406720 sin⁡141−7569408 sin⁡121+7569408 sin⁡101−5406720 sin⁡81+2703360 sin⁡61−901120 sin⁡41+180224 sin⁡21−16384+256139 sin⁡15110240 sin⁡221−112640 sin⁡201+563200 sin⁡181−1689600 sin⁡161+3379200 sin⁡141−4730880 sin⁡121+4730880 sin⁡101−3379200 sin⁡81+1689600 sin⁡61−563200 sin⁡41+112640 sin⁡21−10240+81073 sin⁡712048 sin⁡221−22528 sin⁡201+112640 sin⁡181−337920 sin⁡161+675840 sin⁡141−946176 sin⁡121+946176 sin⁡101−675840 sin⁡81+337920 sin⁡61−112640 sin⁡41+22528 sin⁡21−2048+4199 sin⁡11166 sin⁡221−726 sin⁡201+3630 sin⁡181−10890 sin⁡161+21780 sin⁡141−30492 sin⁡121+30492 sin⁡101−21780 sin⁡81+10890 sin⁡61−3630 sin⁡41+726 sin⁡21−66{{88179\,\log \left(\sin 1+1\right)}\over{1048576}}-{{88179\,\log \left(1-\sin 1\right)}\over{1048576}}-{{22720789\,\sin ^{17}1}\over{ 2621440\,\sin ^{22}1-28835840\,\sin ^{20}1+144179200\,\sin ^{18}1- 432537600\,\sin ^{16}1+865075200\,\sin ^{14}1-1211105280\,\sin ^{12} 1+1211105280\,\sin ^{10}1-865075200\,\sin ^81+432537600\,\sin ^61- 144179200\,\sin ^41+28835840\,\sin ^21-2621440}}-{{37635637\,\sin ^{ 13}1}\over{786432\,\sin ^{22}1-8650752\,\sin ^{20}1+43253760\,\sin ^{18}1-129761280\,\sin ^{16}1+259522560\,\sin ^{14}1-363331584\, \sin ^{12}1+363331584\,\sin ^{10}1-259522560\,\sin ^81+129761280\, \sin ^61-43253760\,\sin ^41+8650752\,\sin ^21-786432}}-{{47036875\, \sin ^91}\over{786432\,\sin ^{22}1-8650752\,\sin ^{20}1+43253760\, \sin ^{18}1-129761280\,\sin ^{16}1+259522560\,\sin ^{14}1-363331584 \,\sin ^{12}1+363331584\,\sin ^{10}1-259522560\,\sin ^81+129761280\, \sin ^61-43253760\,\sin ^41+8650752\,\sin ^21-786432}}-{{88179\, \sin ^{21}1}\over{524288\,\sin ^{22}1-5767168\,\sin ^{20}1+28835840 \,\sin ^{18}1-86507520\,\sin ^{16}1+173015040\,\sin ^{14}1-242221056 \,\sin ^{12}1+242221056\,\sin ^{10}1-173015040\,\sin ^81+86507520\, \sin ^61-28835840\,\sin ^41+5767168\,\sin ^21-524288}}-{{9401903\, \sin ^51}\over{524288\,\sin ^{22}1-5767168\,\sin ^{20}1+28835840\, \sin ^{18}1-86507520\,\sin ^{16}1+173015040\,\sin ^{14}1-242221056\, \sin ^{12}1+242221056\,\sin ^{10}1-173015040\,\sin ^81+86507520\, \sin ^61-28835840\,\sin ^41+5767168\,\sin ^21-524288}}-{{436109\, \sin 1}\over{524288\,\sin ^{22}1-5767168\,\sin ^{20}1+28835840\, \sin ^{18}1-86507520\,\sin ^{16}1+173015040\,\sin ^{14}1-242221056\, \sin ^{12}1+242221056\,\sin ^{10}1-173015040\,\sin ^81+86507520\, \sin ^61-28835840\,\sin ^41+5767168\,\sin ^21-524288}}+{{29393\, \sin ^{19}1}\over{16384\,\sin ^{22}1-180224\,\sin ^{20}1+901120\, \sin ^{18}1-2703360\,\sin ^{16}1+5406720\,\sin ^{14}1-7569408\,\sin ^{12}1+7569408\,\sin ^{10}1-5406720\,\sin ^81+2703360\,\sin ^61- 901120\,\sin ^41+180224\,\sin ^21-16384}}+{{85295\,\sin ^31}\over{ 16384\,\sin ^{22}1-180224\,\sin ^{20}1+901120\,\sin ^{18}1-2703360\, \sin ^{16}1+5406720\,\sin ^{14}1-7569408\,\sin ^{12}1+7569408\,\sin ^{10}1-5406720\,\sin ^81+2703360\,\sin ^61-901120\,\sin ^41+180224\, \sin ^21-16384}}+{{256139\,\sin ^{15}1}\over{10240\,\sin ^{22}1- 112640\,\sin ^{20}1+563200\,\sin ^{18}1-1689600\,\sin ^{16}1+3379200 \,\sin ^{14}1-4730880\,\sin ^{12}1+4730880\,\sin ^{10}1-3379200\, \sin ^81+1689600\,\sin ^61-563200\,\sin ^41+112640\,\sin ^21-10240}} +{{81073\,\sin ^71}\over{2048\,\sin ^{22}1-22528\,\sin ^{20}1+112640 \,\sin ^{18}1-337920\,\sin ^{16}1+675840\,\sin ^{14}1-946176\,\sin ^{12}1+946176\,\sin ^{10}1-675840\,\sin ^81+337920\,\sin ^61-112640 \,\sin ^41+22528\,\sin ^21-2048}}+{{4199\,\sin ^{11}1}\over{66\, \sin ^{22}1-726\,\sin ^{20}1+3630\,\sin ^{18}1-10890\,\sin ^{16}1+ 21780\,\sin ^{14}1-30492\,\sin ^{12}1+30492\,\sin ^{10}1-21780\, \sin ^81+10890\,\sin ^61-3630\,\sin ^41+726\,\sin ^21-66}} 
=
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                                                                             9                    13                    5                   17                                    21                   19                   3                    15                    7                    11             
  88179*log(1 - sin(1))   88179*log(1 + sin(1))              - 5174056250*sin (1) - 4139920070*sin  (1) - 1551313995*sin (1) - 749786037*sin  (1) - 71957985*sin(1) - 14549535*sin  (1) + 155195040*sin  (1) + 450357600*sin (1) + 2163862272*sin  (1) + 3424523520*sin (1) + 5503713280*sin  (1)          
- --------------------- + --------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
         1048576                 1048576                                     12                     8                     16                    4                   20                  22                   2                    18                     6                     14                     10   
                                                  -86507520 - 39966474240*sin  (1) - 28547481600*sin (1) - 14273740800*sin  (1) - 4757913600*sin (1) - 951582720*sin  (1) + 86507520*sin  (1) + 951582720*sin (1) + 4757913600*sin  (1) + 14273740800*sin (1) + 28547481600*sin  (1) + 39966474240*sin  (1)
88179log⁡(sin⁡(1)+1)1048576−88179log⁡(−sin⁡(1)+1)1048576+−5174056250sin⁡9(1)−1551313995sin⁡5(1)−4139920070sin⁡13(1)−71957985sin⁡(1)−749786037sin⁡17(1)−14549535sin⁡21(1)+155195040sin⁡19(1)+2163862272sin⁡15(1)+450357600sin⁡3(1)+5503713280sin⁡11(1)+3424523520sin⁡7(1)−28547481600sin⁡8(1)−39966474240sin⁡12(1)−4757913600sin⁡4(1)−14273740800sin⁡16(1)−86507520−951582720sin⁡20(1)+86507520sin⁡22(1)+4757913600sin⁡18(1)+951582720sin⁡2(1)+28547481600sin⁡14(1)+14273740800sin⁡6(1)+39966474240sin⁡10(1)\frac{88179 \log{\left(\sin{\left(1 \right)} + 1 \right)}}{1048576} - \frac{88179 \log{\left(- \sin{\left(1 \right)} + 1 \right)}}{1048576} + \frac{- 5174056250 \sin^{9}{\left(1 \right)} - 1551313995 \sin^{5}{\left(1 \right)} - 4139920070 \sin^{13}{\left(1 \right)} - 71957985 \sin{\left(1 \right)} - 749786037 \sin^{17}{\left(1 \right)} - 14549535 \sin^{21}{\left(1 \right)} + 155195040 \sin^{19}{\left(1 \right)} + 2163862272 \sin^{15}{\left(1 \right)} + 450357600 \sin^{3}{\left(1 \right)} + 5503713280 \sin^{11}{\left(1 \right)} + 3424523520 \sin^{7}{\left(1 \right)}}{- 28547481600 \sin^{8}{\left(1 \right)} - 39966474240 \sin^{12}{\left(1 \right)} - 4757913600 \sin^{4}{\left(1 \right)} - 14273740800 \sin^{16}{\left(1 \right)} - 86507520 - 951582720 \sin^{20}{\left(1 \right)} + 86507520 \sin^{22}{\left(1 \right)} + 4757913600 \sin^{18}{\left(1 \right)} + 951582720 \sin^{2}{\left(1 \right)} + 28547481600 \sin^{14}{\left(1 \right)} + 14273740800 \sin^{6}{\left(1 \right)} + 39966474240 \sin^{10}{\left(1 \right)}}
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Numerical answer [src] 
42081.8030437732
42081.8030437732
The graph
Integral of 1/(cos²3x) dx

    Use the examples entering the upper and lower limits of integration.

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