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- Integral of d{x}:
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Integral of -e^-x
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Integral of sin(5*x)
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Integral of tan
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Integral of (2x+1)dx
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Identical expressions
- one /(cos²3x)
- 1 divide by ( co sinus of e of ²3x)
- one divide by ( co sinus of e of ²3x)
- 1/cos²3x
- 1 divide by (cos²3x)
- 1/(cos²3x)dx
Integral of 1/(cos²3x) dx
The solution
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[src]
1 / | | 1 | 1*-------- dx | 23 | cos (x) | / 0
$$\int\limits_{0}^{1} 1 \cdot \frac{1}{\cos^{23}{\left(x \right)}}\, dx$$
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) | /
$${{88179\,\log \left(\sin x+1\right)}\over{1048576}}-{{88179\,\log
\left(\sin x-1\right)}\over{1048576}}-{{14549535\,\sin ^{21}x-
155195040\,\sin ^{19}x+749786037\,\sin ^{17}x-2163862272\,\sin ^{15}
x+4139920070\,\sin ^{13}x-5503713280\,\sin ^{11}x+5174056250\,\sin ^
9x-3424523520\,\sin ^7x+1551313995\,\sin ^5x-450357600\,\sin ^3x+
71957985\,\sin x}\over{86507520\,\sin ^{22}x-951582720\,\sin ^{20}x+
4757913600\,\sin ^{18}x-14273740800\,\sin ^{16}x+28547481600\,\sin
^{14}x-39966474240\,\sin ^{12}x+39966474240\,\sin ^{10}x-28547481600
\,\sin ^8x+14273740800\,\sin ^6x-4757913600\,\sin ^4x+951582720\,
\sin ^2x-86507520}}$$
The graph
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 \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}}$$
=
=
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)
$$\frac{88179 \log{\left(\sin{\left(1 \right)} + 1 \right)}}{1048576} - \frac{88179 \log{\left(- \sin{\left(1 \right)} + 1 \right)}}{1048576} + \frac{- 5174056250 \sin^{9}{\left(1 \right)} - 1551313995 \sin^{5}{\left(1 \right)} - 4139920070 \sin^{13}{\left(1 \right)} - 71957985 \sin{\left(1 \right)} - 749786037 \sin^{17}{\left(1 \right)} - 14549535 \sin^{21}{\left(1 \right)} + 155195040 \sin^{19}{\left(1 \right)} + 2163862272 \sin^{15}{\left(1 \right)} + 450357600 \sin^{3}{\left(1 \right)} + 5503713280 \sin^{11}{\left(1 \right)} + 3424523520 \sin^{7}{\left(1 \right)}}{- 28547481600 \sin^{8}{\left(1 \right)} - 39966474240 \sin^{12}{\left(1 \right)} - 4757913600 \sin^{4}{\left(1 \right)} - 14273740800 \sin^{16}{\left(1 \right)} - 86507520 - 951582720 \sin^{20}{\left(1 \right)} + 86507520 \sin^{22}{\left(1 \right)} + 4757913600 \sin^{18}{\left(1 \right)} + 951582720 \sin^{2}{\left(1 \right)} + 28547481600 \sin^{14}{\left(1 \right)} + 14273740800 \sin^{6}{\left(1 \right)} + 39966474240 \sin^{10}{\left(1 \right)}}$$
The graph
Use the examples entering the upper and lower limits of integration.