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Trazado el gráfico de la función/ tan(4*x)/((x*sin(8*x)))

Gráfico de la función y = tan(4*x)/((x*sin(8*x)))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        tan(4*x) 
f(x) = ----------
       x*sin(8*x)
f(x)=tan⁡(4x)xsin⁡(8x)f{\left(x \right)} = \frac{\tan{\left(4 x \right)}}{x \sin{\left(8 x \right)}} 
f = tan(4*x)/((x*sin(8*x)))
Gráfico de la función
02468-8-6-4-2-1010-25002500
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=0.392699081698724x_{2} = 0.392699081698724
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
tan⁡(4x)xsin⁡(8x)=0\frac{\tan{\left(4 x \right)}}{x \sin{\left(8 x \right)}} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en tan(4*x)/((x*sin(8*x))).
tan⁡(0⋅4)0sin⁡(0⋅8)\frac{\tan{\left(0 \cdot 4 \right)}}{0 \sin{\left(0 \cdot 8 \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
1xsin⁡(8x)(4tan⁡2(4x)+4)+(−8xcos⁡(8x)−sin⁡(8x))tan⁡(4x)x2sin⁡2(8x)=0\frac{1}{x \sin{\left(8 x \right)}} \left(4 \tan^{2}{\left(4 x \right)} + 4\right) + \frac{\left(- 8 x \cos{\left(8 x \right)} - \sin{\left(8 x \right)}\right) \tan{\left(4 x \right)}}{x^{2} \sin^{2}{\left(8 x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=69.900883603484x_{1} = 69.900883603484
x2=42.4122376369684x_{2} = 42.4122376369684
x3=−98.1750887333632x_{3} = -98.1750887333632
x4=−95.8189020703721x_{4} = -95.8189020703721
x5=−91.8919251906979x_{5} = -91.8919251906979
x6=21.9925694942943x_{6} = 21.9925694942943
x7=−23.5632711060282x_{7} = -23.5632711060282
x8=82.4671860950413x_{8} = 82.4671860950413
x9=40.0560864868321x_{9} = 40.0560864868321
x10=−21.9925694942943x_{10} = -21.9925694942943
x11=−25.9193450459499x_{11} = -25.9193450459499
x12=94.248111179131x_{12} = 94.248111179131
x13=−10.2132357264933x_{13} = -10.2132357264933
x14=−36.1291804646404x_{14} = -36.1291804646404
x15=−51.8368816356849x_{15} = -51.8368816356849
x16=50.2661041474625x_{16} = 50.2661041474625
x17=72.2570635158036x_{17} = 72.2570635158036
x18=14.1393770233852x_{18} = 14.1393770233852
x19=20.421882450548x_{19} = 20.421882450548
x20=−55.7638299993776x_{20} = -55.7638299993776
x21=62.046958558512x_{21} = 62.046958558512
x22=38.4853220015028x_{22} = 38.4853220015028
x23=32.2022951221465x_{23} = 32.2022951221465
x24=−27.4900724850885x_{24} = -27.4900724850885
x25=16.4952558791847x_{25} = 16.4952558791847
x26=29.8461772382431x_{26} = 29.8461772382431
x27=−73.827850640952x_{27} = -73.827850640952
x28=−40.0560864868321x_{28} = -40.0560864868321
x29=18.065887509935x_{29} = 18.065887509935
x30=73.827850640952x_{30} = 73.827850640952
x31=46.3391660143515x_{31} = 46.3391660143515
x32=−32.2022951221465x_{32} = -32.2022951221465
x33=−62.046958558512x_{33} = -62.046958558512
x34=87.9649495554097x_{34} = 87.9649495554097
x35=−77.7548200803457x_{35} = -77.7548200803457
x36=−1.59040509801642x_{36} = -1.59040509801642
x37=−58.1200017712265x_{37} = -58.1200017712265
x38=−69.900883603484x_{38} = -69.900883603484
x39=−71.4716701049018x_{39} = -71.4716701049018
x40=−41.626853375133x_{40} = -41.626853375133
x41=80.1110027499685x_{41} = 80.1110027499685
x42=−87.9649495554097x_{42} = -87.9649495554097
x43=−7.85795815864751x_{43} = -7.85795815864751
x44=54.19304991561x_{44} = 54.19304991561
x45=−3.93492983900122x_{45} = -3.93492983900122
x46=95.8189020703721x_{46} = 95.8189020703721
x47=−54.19304991561x_{47} = -54.19304991561
x48=43.9830076500054x_{48} = 43.9830076500054
x49=51.8368816356849x_{49} = 51.8368816356849
x50=60.4761753132807x_{50} = 60.4761753132807
x51=−33.7730463159505x_{51} = -33.7730463159505
x52=90.321134778167x_{52} = 90.321134778167
x53=−81.6817915752418x_{53} = -81.6817915752418
x54=−14.1393770233852x_{54} = -14.1393770233852
x55=86.3941596877322x_{55} = 86.3941596877322
x56=−49.4807158518691x_{56} = -49.4807158518691
x57=−65.9739193968439x_{57} = -65.9739193968439
x58=76.1840320401281x_{58} = 76.1840320401281
x59=33.7730463159505x_{59} = 33.7730463159505
x60=−59.690783948846x_{60} = -59.690783948846
x61=77.7548200803457x_{61} = 77.7548200803457
x62=28.2754390745685x_{62} = 28.2754390745685
x63=−5.50346440915574x_{63} = -5.50346440915574
x64=100.53127576325x_{64} = 100.53127576325
x65=11.7836243369193x_{65} = 11.7836243369193
x66=84.037975338951x_{66} = 84.037975338951
x67=2.3693714263552x_{67} = 2.3693714263552
x68=47.9099402312409x_{68} = 47.9099402312409
x69=−67.5447047082016x_{69} = -67.5447047082016
x70=−84.037975338951x_{70} = -84.037975338951
x71=−11.7836243369193x_{71} = -11.7836243369193
x72=55.7638299993776x_{72} = 55.7638299993776
x73=6.2881543144839x_{73} = 6.2881543144839
x74=−99.7458800474599x_{74} = -99.7458800474599
x75=7.85795815864751x_{75} = 7.85795815864751
x76=−29.8461772382431x_{76} = -29.8461772382431
x77=−15.709952410865x_{77} = -15.709952410865
x78=98.1750887333632x_{78} = 98.1750887333632
x79=−85.6087648427791x_{79} = -85.6087648427791
x80=24.3486264939971x_{80} = 24.3486264939971
x81=25.9193450459499x_{81} = 25.9193450459499
x82=58.1200017712265x_{82} = 58.1200017712265
x83=3.93492983900122x_{83} = 3.93492983900122
x84=68.3300975537973x_{84} = 68.3300975537973
x85=−63.6177424497285x_{85} = -63.6177424497285
x86=65.9739193968439x_{86} = 65.9739193968439
x87=10.2132357264933x_{87} = 10.2132357264933
x88=−45.5537794776852x_{88} = -45.5537794776852
x89=−89.5357396497198x_{89} = -89.5357396497198
x90=−94.248111179131x_{90} = -94.248111179131
x91=−43.9830076500054x_{91} = -43.9830076500054
x92=−19.6365454838884x_{92} = -19.6365454838884
x93=−18.065887509935x_{93} = -18.065887509935
x94=36.1291804646404x_{94} = 36.1291804646404
x95=−47.9099402312409x_{95} = -47.9099402312409
x96=−37.6999407538095x_{96} = -37.6999407538095
x97=91.8919251906979x_{97} = 91.8919251906979
x98=−80.1110027499685x_{98} = -80.1110027499685
x99=64.4031346228107x_{99} = 64.4031346228107
x100=−76.1840320401281x_{100} = -76.1840320401281
Signos de extremos en los puntos:
(69.90088360348398, 0.0071530082759581)

(42.41223763696837, 0.0117891526369848)

(-98.17508873336317, -0.00509294992258353)

(-95.81890207037209, -0.00521818597494121)

(-91.89192519069788, -0.00544118456721579)

(21.99256949429425, 0.0227356859128165)

(-23.563271106028246, -0.0212200618731021)

(82.46718609504133, 0.00606303152116114)

(40.05608648683205, 0.0124826190724505)

(-21.99256949429425, -0.0227356859128165)

(-25.919345045949925, -0.0192910595581496)

(94.24811117913099, 0.00530515543773241)

(-10.213235726493343, -0.0489634147494005)

(-36.1291804646404, -0.0138393946029972)

(-51.83688163568488, -0.00964569803724581)

(50.26610414746248, 0.00994712242933777)

(72.25706351580364, 0.00691975942570154)

(14.139377023385192, 0.0353650006585651)

(20.42188245054795, 0.0244844584640621)

(-55.76382999937763, -0.00896643061250936)

(62.04695855851195, 0.00805844542470958)

(38.48532200150282, 0.0129921031894958)

(32.202295122146545, 0.0155270775559065)

(-27.490072485088508, -0.0181887602629081)

(16.4952558791847, 0.0303134862667421)

(29.846177238243104, 0.0167528580384504)

(-73.82785064095195, -0.00677253135504176)

(-40.05608648683205, -0.0124826190724505)

(18.065887509935, 0.0276777953377112)

(73.82785064095195, 0.00677253135504176)

(46.33916601435151, 0.0107900871177542)

(-32.202295122146545, -0.0155270775559065)

(-62.04695855851195, -0.00805844542470958)

(87.96494955540975, 0.00568409363248406)

(-77.7548200803457, -0.00643048613192283)

(-1.5904050980164162, -0.316327388524221)

(-58.12000177122646, -0.0086029301026508)

(-69.90088360348398, -0.0071530082759581)

(-71.47167010490178, -0.00699580027540903)

(-41.626853375132974, -0.0120115855048199)

(80.11100274996855, 0.00624135512175728)

(-87.96494955540975, -0.00568409363248406)

(-7.857958158647509, -0.0636458623513395)

(54.19304991561002, 0.00922632442552143)

(-3.9349298390012173, -0.127195295526552)

(95.81890207037209, 0.00521818597494121)

(-54.19304991561002, -0.00922632442552143)

(43.98300765000541, 0.0113681183988067)

(51.83688163568488, 0.00964569803724581)

(60.476175313280685, 0.00826775392969158)

(-33.77304631595054, -0.014804908170721)

(90.32113477816702, 0.00553581350464464)

(-81.68179157524183, -0.00612132962941409)

(-14.139377023385192, -0.0353650006585651)

(86.39415968773224, 0.00578744036060897)

(-49.480715851869085, -0.0101050112620193)

(-65.97391939684388, -0.00757877960704299)

(76.18403204012813, 0.00656307276818217)

(33.77304631595054, 0.014804908170721)

(-59.69078394884601, -0.00837653921767138)

(77.7548200803457, 0.00643048613192283)

(28.275439074568542, 0.017683536952623)

(-5.5034644091557405, -0.0908987325474447)

(100.53127576325025, 0.00497358428228842)

(11.783624336919265, 0.0424365415865174)

(84.037975338951, 0.00594970433541056)

(2.369371426355199, 0.211613774133006)

(47.909940231240945, 0.010436318667665)

(-67.54470470820162, -0.00740253014013025)

(-84.037975338951, -0.00594970433541056)

(-11.783624336919265, -0.0424365415865174)

(55.76382999937763, 0.00896643061250936)

(6.288154314483904, 0.0795460090970505)

(-99.74588004745995, -0.005012746240726)

(7.857958158647509, 0.0636458623513395)

(-29.846177238243104, -0.0167528580384504)

(-15.709952410864979, -0.031828973237785)

(98.17508873336317, 0.00509294992258353)

(-85.60876484277911, -0.00584053591836453)

(24.348626493997077, 0.020535580430713)

(25.919345045949925, 0.0192910595581496)

(58.12000177122646, 0.0086029301026508)

(3.9349298390012173, 0.127195295526552)

(68.33009755379732, 0.00731744415959671)

(-63.61774244972846, -0.00785947301939915)

(65.97391939684388, 0.00757877960704299)

(10.213235726493343, 0.0489634147494005)

(-45.55377947768522, -0.0109761203247423)

(-89.53573964971982, -0.0055843730837597)

(-94.24811117913099, -0.00530515543773241)

(-43.98300765000541, -0.0113681183988067)

(-19.636545483888366, -0.0254637589571234)

(-18.065887509935, -0.0276777953377112)

(36.1291804646404, 0.0138393946029972)

(-47.909940231240945, -0.010436318667665)

(-37.699940753809464, -0.0132627661154553)

(91.89192519069788, 0.00544118456721579)

(-80.11100274996855, -0.00624135512175728)

(64.40313462281071, 0.00776362651404215)

(-76.18403204012813, -0.00656307276818217)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=69.900883603484x_{1} = 69.900883603484
x2=42.4122376369684x_{2} = 42.4122376369684
x3=21.9925694942943x_{3} = 21.9925694942943
x4=82.4671860950413x_{4} = 82.4671860950413
x5=40.0560864868321x_{5} = 40.0560864868321
x6=94.248111179131x_{6} = 94.248111179131
x7=50.2661041474625x_{7} = 50.2661041474625
x8=72.2570635158036x_{8} = 72.2570635158036
x9=14.1393770233852x_{9} = 14.1393770233852
x10=20.421882450548x_{10} = 20.421882450548
x11=62.046958558512x_{11} = 62.046958558512
x12=38.4853220015028x_{12} = 38.4853220015028
x13=32.2022951221465x_{13} = 32.2022951221465
x14=16.4952558791847x_{14} = 16.4952558791847
x15=29.8461772382431x_{15} = 29.8461772382431
x16=18.065887509935x_{16} = 18.065887509935
x17=73.827850640952x_{17} = 73.827850640952
x18=46.3391660143515x_{18} = 46.3391660143515
x19=87.9649495554097x_{19} = 87.9649495554097
x20=80.1110027499685x_{20} = 80.1110027499685
x21=54.19304991561x_{21} = 54.19304991561
x22=95.8189020703721x_{22} = 95.8189020703721
x23=43.9830076500054x_{23} = 43.9830076500054
x24=51.8368816356849x_{24} = 51.8368816356849
x25=60.4761753132807x_{25} = 60.4761753132807
x26=90.321134778167x_{26} = 90.321134778167
x27=86.3941596877322x_{27} = 86.3941596877322
x28=76.1840320401281x_{28} = 76.1840320401281
x29=33.7730463159505x_{29} = 33.7730463159505
x30=77.7548200803457x_{30} = 77.7548200803457
x31=28.2754390745685x_{31} = 28.2754390745685
x32=100.53127576325x_{32} = 100.53127576325
x33=11.7836243369193x_{33} = 11.7836243369193
x34=84.037975338951x_{34} = 84.037975338951
x35=2.3693714263552x_{35} = 2.3693714263552
x36=47.9099402312409x_{36} = 47.9099402312409
x37=55.7638299993776x_{37} = 55.7638299993776
x38=6.2881543144839x_{38} = 6.2881543144839
x39=7.85795815864751x_{39} = 7.85795815864751
x40=98.1750887333632x_{40} = 98.1750887333632
x41=24.3486264939971x_{41} = 24.3486264939971
x42=25.9193450459499x_{42} = 25.9193450459499
x43=58.1200017712265x_{43} = 58.1200017712265
x44=3.93492983900122x_{44} = 3.93492983900122
x45=68.3300975537973x_{45} = 68.3300975537973
x46=65.9739193968439x_{46} = 65.9739193968439
x47=10.2132357264933x_{47} = 10.2132357264933
x48=36.1291804646404x_{48} = 36.1291804646404
x49=91.8919251906979x_{49} = 91.8919251906979
x50=64.4031346228107x_{50} = 64.4031346228107
Puntos máximos de la función:
x50=−98.1750887333632x_{50} = -98.1750887333632
x50=−95.8189020703721x_{50} = -95.8189020703721
x50=−91.8919251906979x_{50} = -91.8919251906979
x50=−23.5632711060282x_{50} = -23.5632711060282
x50=−21.9925694942943x_{50} = -21.9925694942943
x50=−25.9193450459499x_{50} = -25.9193450459499
x50=−10.2132357264933x_{50} = -10.2132357264933
x50=−36.1291804646404x_{50} = -36.1291804646404
x50=−51.8368816356849x_{50} = -51.8368816356849
x50=−55.7638299993776x_{50} = -55.7638299993776
x50=−27.4900724850885x_{50} = -27.4900724850885
x50=−73.827850640952x_{50} = -73.827850640952
x50=−40.0560864868321x_{50} = -40.0560864868321
x50=−32.2022951221465x_{50} = -32.2022951221465
x50=−62.046958558512x_{50} = -62.046958558512
x50=−77.7548200803457x_{50} = -77.7548200803457
x50=−1.59040509801642x_{50} = -1.59040509801642
x50=−58.1200017712265x_{50} = -58.1200017712265
x50=−69.900883603484x_{50} = -69.900883603484
x50=−71.4716701049018x_{50} = -71.4716701049018
x50=−41.626853375133x_{50} = -41.626853375133
x50=−87.9649495554097x_{50} = -87.9649495554097
x50=−7.85795815864751x_{50} = -7.85795815864751
x50=−3.93492983900122x_{50} = -3.93492983900122
x50=−54.19304991561x_{50} = -54.19304991561
x50=−33.7730463159505x_{50} = -33.7730463159505
x50=−81.6817915752418x_{50} = -81.6817915752418
x50=−14.1393770233852x_{50} = -14.1393770233852
x50=−49.4807158518691x_{50} = -49.4807158518691
x50=−65.9739193968439x_{50} = -65.9739193968439
x50=−59.690783948846x_{50} = -59.690783948846
x50=−5.50346440915574x_{50} = -5.50346440915574
x50=−67.5447047082016x_{50} = -67.5447047082016
x50=−84.037975338951x_{50} = -84.037975338951
x50=−11.7836243369193x_{50} = -11.7836243369193
x50=−99.7458800474599x_{50} = -99.7458800474599
x50=−29.8461772382431x_{50} = -29.8461772382431
x50=−15.709952410865x_{50} = -15.709952410865
x50=−85.6087648427791x_{50} = -85.6087648427791
x50=−63.6177424497285x_{50} = -63.6177424497285
x50=−45.5537794776852x_{50} = -45.5537794776852
x50=−89.5357396497198x_{50} = -89.5357396497198
x50=−94.248111179131x_{50} = -94.248111179131
x50=−43.9830076500054x_{50} = -43.9830076500054
x50=−19.6365454838884x_{50} = -19.6365454838884
x50=−18.065887509935x_{50} = -18.065887509935
x50=−47.9099402312409x_{50} = -47.9099402312409
x50=−37.6999407538095x_{50} = -37.6999407538095
x50=−80.1110027499685x_{50} = -80.1110027499685
x50=−76.1840320401281x_{50} = -76.1840320401281
Decrece en los intervalos
[100.53127576325,∞)\left[100.53127576325, \infty\right)
Crece en los intervalos
[−1.59040509801642,2.3693714263552]\left[-1.59040509801642, 2.3693714263552\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
32(tan⁡2(4x)+1)tan⁡(4x)−8(8xcos⁡(8x)+sin⁡(8x))(tan⁡2(4x)+1)xsin⁡(8x)+(64xsin⁡(8x)+(8xcos⁡(8x)+sin⁡(8x))(8cos⁡(8x)sin⁡(8x)+1x)+8(8xcos⁡(8x)+sin⁡(8x))cos⁡(8x)sin⁡(8x)−16cos⁡(8x)+8xcos⁡(8x)+sin⁡(8x)x)tan⁡(4x)xsin⁡(8x)xsin⁡(8x)=0\frac{32 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} - \frac{8 \left(8 x \cos{\left(8 x \right)} + \sin{\left(8 x \right)}\right) \left(\tan^{2}{\left(4 x \right)} + 1\right)}{x \sin{\left(8 x \right)}} + \frac{\left(64 x \sin{\left(8 x \right)} + \left(8 x \cos{\left(8 x \right)} + \sin{\left(8 x \right)}\right) \left(\frac{8 \cos{\left(8 x \right)}}{\sin{\left(8 x \right)}} + \frac{1}{x}\right) + \frac{8 \left(8 x \cos{\left(8 x \right)} + \sin{\left(8 x \right)}\right) \cos{\left(8 x \right)}}{\sin{\left(8 x \right)}} - 16 \cos{\left(8 x \right)} + \frac{8 x \cos{\left(8 x \right)} + \sin{\left(8 x \right)}}{x}\right) \tan{\left(4 x \right)}}{x \sin{\left(8 x \right)}}}{x \sin{\left(8 x \right)}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=0.392699081698724x_{2} = 0.392699081698724
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(tan⁡(4x)xsin⁡(8x))y = \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)}}{x \sin{\left(8 x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(4x)xsin⁡(8x))y = \lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)}}{x \sin{\left(8 x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(4*x)/((x*sin(8*x))), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(1xsin⁡(8x)tan⁡(4x)x)y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{x \sin{\left(8 x \right)}} \tan{\left(4 x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(1xsin⁡(8x)tan⁡(4x)x)y = x \lim_{x \to \infty}\left(\frac{\frac{1}{x \sin{\left(8 x \right)}} \tan{\left(4 x \right)}}{x}\right)
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
tan⁡(4x)xsin⁡(8x)=−1xsin⁡(8x)tan⁡(4x)\frac{\tan{\left(4 x \right)}}{x \sin{\left(8 x \right)}} = - \frac{1}{x \sin{\left(8 x \right)}} \tan{\left(4 x \right)}
- No
tan⁡(4x)xsin⁡(8x)=1xsin⁡(8x)tan⁡(4x)\frac{\tan{\left(4 x \right)}}{x \sin{\left(8 x \right)}} = \frac{1}{x \sin{\left(8 x \right)}} \tan{\left(4 x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор