Sr Examen
Integral de 1÷sin²3x dx
Solución
Ha introducido
[src]
pi -- 9 / | | 1 | -------- dx | 23 | sin (x) | / pi -- 12
$$\int\limits_{\frac{\pi}{12}}^{\frac{\pi}{9}} \frac{1}{\sin^{23}{\left(x \right)}}\, dx$$
Integral(1/(sin(x)^23), (x, pi/12, pi/9))
Respuesta (Indefinida)
[src]
/ | 11 7 15 3 19 21 17 5 13 9 | 1 88179*log(1 + cos(x)) 88179*log(-1 + cos(x)) - 5503713280*cos (x) - 3424523520*cos (x) - 2163862272*cos (x) - 450357600*cos (x) - 155195040*cos (x) + 14549535*cos (x) + 71957985*cos(x) + 749786037*cos (x) + 1551313995*cos (x) + 4139920070*cos (x) + 5174056250*cos (x) | -------- dx = C - --------------------- + ---------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | 23 1048576 1048576 12 8 16 4 20 22 2 18 6 14 10 | sin (x) -86507520 - 39966474240*cos (x) - 28547481600*cos (x) - 14273740800*cos (x) - 4757913600*cos (x) - 951582720*cos (x) + 86507520*cos (x) + 951582720*cos (x) + 4757913600*cos (x) + 14273740800*cos (x) + 28547481600*cos (x) + 39966474240*cos (x) | /
$$\int \frac{1}{\sin^{23}{\left(x \right)}}\, dx = C + \frac{14549535 \cos^{21}{\left(x \right)} - 155195040 \cos^{19}{\left(x \right)} + 749786037 \cos^{17}{\left(x \right)} - 2163862272 \cos^{15}{\left(x \right)} + 4139920070 \cos^{13}{\left(x \right)} - 5503713280 \cos^{11}{\left(x \right)} + 5174056250 \cos^{9}{\left(x \right)} - 3424523520 \cos^{7}{\left(x \right)} + 1551313995 \cos^{5}{\left(x \right)} - 450357600 \cos^{3}{\left(x \right)} + 71957985 \cos{\left(x \right)}}{86507520 \cos^{22}{\left(x \right)} - 951582720 \cos^{20}{\left(x \right)} + 4757913600 \cos^{18}{\left(x \right)} - 14273740800 \cos^{16}{\left(x \right)} + 28547481600 \cos^{14}{\left(x \right)} - 39966474240 \cos^{12}{\left(x \right)} + 39966474240 \cos^{10}{\left(x \right)} - 28547481600 \cos^{8}{\left(x \right)} + 14273740800 \cos^{6}{\left(x \right)} - 4757913600 \cos^{4}{\left(x \right)} + 951582720 \cos^{2}{\left(x \right)} - 86507520} + \frac{88179 \log{\left(\cos{\left(x \right)} - 1 \right)}}{1048576} - \frac{88179 \log{\left(\cos{\left(x \right)} + 1 \right)}}{1048576}$$
Gráfica
Respuesta
[src]
11 7 15 3 19 21 17 5 13 9
/ ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ ___ ___
/ /pi\\ | \/ 2 \/ 6 | / /pi\\ | \/ 2 \/ 6 | 11/pi\ 7/pi\ 15/pi\ 3/pi\ 19/pi\ 21/pi\ /pi\ 17/pi\ 5/pi\ 13/pi\ 9/pi\ |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | 71957985*\/ 2 71957985*\/ 6
88179*log|1 + cos|--|| 88179*log|1 - ----- - -----| 88179*log|1 - cos|--|| 88179*log|1 + ----- + -----| - 5503713280*cos |--| - 3424523520*cos |--| - 2163862272*cos |--| - 450357600*cos |--| - 155195040*cos |--| + 14549535*cos |--| + 71957985*cos|--| + 749786037*cos |--| + 1551313995*cos |--| + 4139920070*cos |--| + 5174056250*cos |--| - 5503713280*|----- + -----| - 3424523520*|----- + -----| - 2163862272*|----- + -----| - 450357600*|----- + -----| - 155195040*|----- + -----| + 14549535*|----- + -----| + 749786037*|----- + -----| + 1551313995*|----- + -----| + 4139920070*|----- + -----| + 5174056250*|----- + -----| + -------------- + --------------
\ \9 // \ 4 4 / \ \9 // \ 4 4 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / 4 4
- ---------------------- - ---------------------------- + ---------------------- + ---------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1048576 1048576 1048576 1048576 12/pi\ 8/pi\ 16/pi\ 4/pi\ 20/pi\ 22/pi\ 2/pi\ 18/pi\ 6/pi\ 14/pi\ 10/pi\ 12 8 16 4 20 22 2 18 6 14 10
-86507520 - 39966474240*cos |--| - 28547481600*cos |--| - 14273740800*cos |--| - 4757913600*cos |--| - 951582720*cos |--| + 86507520*cos |--| + 951582720*cos |--| + 4757913600*cos |--| + 14273740800*cos |--| + 28547481600*cos |--| + 39966474240*cos |--| / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\
\9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 |
-86507520 - 39966474240*|----- + -----| - 28547481600*|----- + -----| - 14273740800*|----- + -----| - 4757913600*|----- + -----| - 951582720*|----- + -----| + 86507520*|----- + -----| + 951582720*|----- + -----| + 4757913600*|----- + -----| + 14273740800*|----- + -----| + 28547481600*|----- + -----| + 39966474240*|----- + -----|
\ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 /
$$\frac{- 5503713280 \cos^{11}{\left(\frac{\pi}{9} \right)} - 3424523520 \cos^{7}{\left(\frac{\pi}{9} \right)} - 2163862272 \cos^{15}{\left(\frac{\pi}{9} \right)} - 450357600 \cos^{3}{\left(\frac{\pi}{9} \right)} - 155195040 \cos^{19}{\left(\frac{\pi}{9} \right)} + 14549535 \cos^{21}{\left(\frac{\pi}{9} \right)} + 71957985 \cos{\left(\frac{\pi}{9} \right)} + 749786037 \cos^{17}{\left(\frac{\pi}{9} \right)} + 1551313995 \cos^{5}{\left(\frac{\pi}{9} \right)} + 4139920070 \cos^{13}{\left(\frac{\pi}{9} \right)} + 5174056250 \cos^{9}{\left(\frac{\pi}{9} \right)}}{- 39966474240 \cos^{12}{\left(\frac{\pi}{9} \right)} - 28547481600 \cos^{8}{\left(\frac{\pi}{9} \right)} - 14273740800 \cos^{16}{\left(\frac{\pi}{9} \right)} - 4757913600 \cos^{4}{\left(\frac{\pi}{9} \right)} - 951582720 \cos^{20}{\left(\frac{\pi}{9} \right)} - 86507520 + 86507520 \cos^{22}{\left(\frac{\pi}{9} \right)} + 951582720 \cos^{2}{\left(\frac{\pi}{9} \right)} + 4757913600 \cos^{18}{\left(\frac{\pi}{9} \right)} + 14273740800 \cos^{6}{\left(\frac{\pi}{9} \right)} + 28547481600 \cos^{14}{\left(\frac{\pi}{9} \right)} + 39966474240 \cos^{10}{\left(\frac{\pi}{9} \right)}} + \frac{88179 \log{\left(1 - \cos{\left(\frac{\pi}{9} \right)} \right)}}{1048576} - \frac{88179 \log{\left(\cos{\left(\frac{\pi}{9} \right)} + 1 \right)}}{1048576} + \frac{88179 \log{\left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} + 1 \right)}}{1048576} - \frac{88179 \log{\left(- \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} + 1 \right)}}{1048576} - \frac{- 5503713280 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{11} - 3424523520 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{7} - 2163862272 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{15} - 450357600 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{3} - 155195040 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{19} + 14549535 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{21} + \frac{71957985 \sqrt{2}}{4} + \frac{71957985 \sqrt{6}}{4} + 749786037 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{17} + 1551313995 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{5} + 4139920070 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{13} + 5174056250 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{9}}{- 39966474240 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{12} - 28547481600 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{8} - 14273740800 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{16} - 4757913600 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{4} - 951582720 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{20} - 86507520 + 86507520 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{22} + 951582720 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{2} + 4757913600 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{18} + 14273740800 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{6} + 28547481600 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{14} + 39966474240 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{10}}$$
=
=
11 7 15 3 19 21 17 5 13 9
/ ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ ___ ___
/ /pi\\ | \/ 2 \/ 6 | / /pi\\ | \/ 2 \/ 6 | 11/pi\ 7/pi\ 15/pi\ 3/pi\ 19/pi\ 21/pi\ /pi\ 17/pi\ 5/pi\ 13/pi\ 9/pi\ |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | 71957985*\/ 2 71957985*\/ 6
88179*log|1 + cos|--|| 88179*log|1 - ----- - -----| 88179*log|1 - cos|--|| 88179*log|1 + ----- + -----| - 5503713280*cos |--| - 3424523520*cos |--| - 2163862272*cos |--| - 450357600*cos |--| - 155195040*cos |--| + 14549535*cos |--| + 71957985*cos|--| + 749786037*cos |--| + 1551313995*cos |--| + 4139920070*cos |--| + 5174056250*cos |--| - 5503713280*|----- + -----| - 3424523520*|----- + -----| - 2163862272*|----- + -----| - 450357600*|----- + -----| - 155195040*|----- + -----| + 14549535*|----- + -----| + 749786037*|----- + -----| + 1551313995*|----- + -----| + 4139920070*|----- + -----| + 5174056250*|----- + -----| + -------------- + --------------
\ \9 // \ 4 4 / \ \9 // \ 4 4 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / 4 4
- ---------------------- - ---------------------------- + ---------------------- + ---------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1048576 1048576 1048576 1048576 12/pi\ 8/pi\ 16/pi\ 4/pi\ 20/pi\ 22/pi\ 2/pi\ 18/pi\ 6/pi\ 14/pi\ 10/pi\ 12 8 16 4 20 22 2 18 6 14 10
-86507520 - 39966474240*cos |--| - 28547481600*cos |--| - 14273740800*cos |--| - 4757913600*cos |--| - 951582720*cos |--| + 86507520*cos |--| + 951582720*cos |--| + 4757913600*cos |--| + 14273740800*cos |--| + 28547481600*cos |--| + 39966474240*cos |--| / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\
\9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / \9 / |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 | |\/ 2 \/ 6 |
-86507520 - 39966474240*|----- + -----| - 28547481600*|----- + -----| - 14273740800*|----- + -----| - 4757913600*|----- + -----| - 951582720*|----- + -----| + 86507520*|----- + -----| + 951582720*|----- + -----| + 4757913600*|----- + -----| + 14273740800*|----- + -----| + 28547481600*|----- + -----| + 39966474240*|----- + -----|
\ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 / \ 4 4 /
$$\frac{- 5503713280 \cos^{11}{\left(\frac{\pi}{9} \right)} - 3424523520 \cos^{7}{\left(\frac{\pi}{9} \right)} - 2163862272 \cos^{15}{\left(\frac{\pi}{9} \right)} - 450357600 \cos^{3}{\left(\frac{\pi}{9} \right)} - 155195040 \cos^{19}{\left(\frac{\pi}{9} \right)} + 14549535 \cos^{21}{\left(\frac{\pi}{9} \right)} + 71957985 \cos{\left(\frac{\pi}{9} \right)} + 749786037 \cos^{17}{\left(\frac{\pi}{9} \right)} + 1551313995 \cos^{5}{\left(\frac{\pi}{9} \right)} + 4139920070 \cos^{13}{\left(\frac{\pi}{9} \right)} + 5174056250 \cos^{9}{\left(\frac{\pi}{9} \right)}}{- 39966474240 \cos^{12}{\left(\frac{\pi}{9} \right)} - 28547481600 \cos^{8}{\left(\frac{\pi}{9} \right)} - 14273740800 \cos^{16}{\left(\frac{\pi}{9} \right)} - 4757913600 \cos^{4}{\left(\frac{\pi}{9} \right)} - 951582720 \cos^{20}{\left(\frac{\pi}{9} \right)} - 86507520 + 86507520 \cos^{22}{\left(\frac{\pi}{9} \right)} + 951582720 \cos^{2}{\left(\frac{\pi}{9} \right)} + 4757913600 \cos^{18}{\left(\frac{\pi}{9} \right)} + 14273740800 \cos^{6}{\left(\frac{\pi}{9} \right)} + 28547481600 \cos^{14}{\left(\frac{\pi}{9} \right)} + 39966474240 \cos^{10}{\left(\frac{\pi}{9} \right)}} + \frac{88179 \log{\left(1 - \cos{\left(\frac{\pi}{9} \right)} \right)}}{1048576} - \frac{88179 \log{\left(\cos{\left(\frac{\pi}{9} \right)} + 1 \right)}}{1048576} + \frac{88179 \log{\left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} + 1 \right)}}{1048576} - \frac{88179 \log{\left(- \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} + 1 \right)}}{1048576} - \frac{- 5503713280 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{11} - 3424523520 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{7} - 2163862272 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{15} - 450357600 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{3} - 155195040 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{19} + 14549535 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{21} + \frac{71957985 \sqrt{2}}{4} + \frac{71957985 \sqrt{6}}{4} + 749786037 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{17} + 1551313995 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{5} + 4139920070 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{13} + 5174056250 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{9}}{- 39966474240 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{12} - 28547481600 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{8} - 14273740800 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{16} - 4757913600 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{4} - 951582720 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{20} - 86507520 + 86507520 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{22} + 951582720 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{2} + 4757913600 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{18} + 14273740800 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{6} + 28547481600 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{14} + 39966474240 \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)^{10}}$$
-88179*log(1 + cos(pi/9))/1048576 - 88179*log(1 - sqrt(2)/4 - sqrt(6)/4)/1048576 + 88179*log(1 - cos(pi/9))/1048576 + 88179*log(1 + sqrt(2)/4 + sqrt(6)/4)/1048576 + (-5503713280*cos(pi/9)^11 - 3424523520*cos(pi/9)^7 - 2163862272*cos(pi/9)^15 - 450357600*cos(pi/9)^3 - 155195040*cos(pi/9)^19 + 14549535*cos(pi/9)^21 + 71957985*cos(pi/9) + 749786037*cos(pi/9)^17 + 1551313995*cos(pi/9)^5 + 4139920070*cos(pi/9)^13 + 5174056250*cos(pi/9)^9)/(-86507520 - 39966474240*cos(pi/9)^12 - 28547481600*cos(pi/9)^8 - 14273740800*cos(pi/9)^16 - 4757913600*cos(pi/9)^4 - 951582720*cos(pi/9)^20 + 86507520*cos(pi/9)^22 + 951582720*cos(pi/9)^2 + 4757913600*cos(pi/9)^18 + 14273740800*cos(pi/9)^6 + 28547481600*cos(pi/9)^14 + 39966474240*cos(pi/9)^10) - (-5503713280*(sqrt(2)/4 + sqrt(6)/4)^11 - 3424523520*(sqrt(2)/4 + sqrt(6)/4)^7 - 2163862272*(sqrt(2)/4 + sqrt(6)/4)^15 - 450357600*(sqrt(2)/4 + sqrt(6)/4)^3 - 155195040*(sqrt(2)/4 + sqrt(6)/4)^19 + 14549535*(sqrt(2)/4 + sqrt(6)/4)^21 + 749786037*(sqrt(2)/4 + sqrt(6)/4)^17 + 1551313995*(sqrt(2)/4 + sqrt(6)/4)^5 + 4139920070*(sqrt(2)/4 + sqrt(6)/4)^13 + 5174056250*(sqrt(2)/4 + sqrt(6)/4)^9 + 71957985*sqrt(2)/4 + 71957985*sqrt(6)/4)/(-86507520 - 39966474240*(sqrt(2)/4 + sqrt(6)/4)^12 - 28547481600*(sqrt(2)/4 + sqrt(6)/4)^8 - 14273740800*(sqrt(2)/4 + sqrt(6)/4)^16 - 4757913600*(sqrt(2)/4 + sqrt(6)/4)^4 - 951582720*(sqrt(2)/4 + sqrt(6)/4)^20 + 86507520*(sqrt(2)/4 + sqrt(6)/4)^22 + 951582720*(sqrt(2)/4 + sqrt(6)/4)^2 + 4757913600*(sqrt(2)/4 + sqrt(6)/4)^18 + 14273740800*(sqrt(2)/4 + sqrt(6)/4)^6 + 28547481600*(sqrt(2)/4 + sqrt(6)/4)^14 + 39966474240*(sqrt(2)/4 + sqrt(6)/4)^10)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.