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- Integrales paso a paso
- Gráfico de la función paramétrica
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- Superficie especificada paramétricamente
- Superficie dada por la ecuación
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- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^3+3*x^2+2
-
-sqrt(3*x+1)
-
x*(-1-log(x))
-
-cos(2*x)-sin(2*x)
-
Expresiones idénticas
- sinx/(x^ dos +x)
- seno de x dividir por (x al cuadrado más x)
- seno de x dividir por (x en el grado dos más x)
- sinx/(x2+x)
- sinx/x2+x
- sinx/(x²+x)
- sinx/(x en el grado 2+x)
- sinx/x^2+x
- sinx dividir por (x^2+x)
-
Expresiones semejantes
- sinx/(x^2-x)
-
Expresiones con funciones
- sinx
- sinx-log(e,sin(x))
- sinx+sin^5(2x)
- sinx/(1-x)
- sinx/(1-x^(sin2x))
- sinx/|x|
Gráfico de la función y = sinx/(x^2+x)
Solución
Ha introducido
[src]
sin(x)
f(x) = ------
2
x + x
$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{x^{2} + x}$$
f = sin(x)/(x^2 + x)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = -1$$
$$x_{2} = 0$$
$$x_{1} = -1$$
$$x_{2} = 0$$
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:
$$\frac{\sin{\left(x \right)}}{x^{2} + x} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \pi$$
Solución numérica
$$x_{1} = 40.8407044966673$$
$$x_{2} = -135.088484104361$$
$$x_{3} = -18.8495559215388$$
$$x_{4} = -56.5486677646163$$
$$x_{5} = 97.3893722612836$$
$$x_{6} = 34.5575191894877$$
$$x_{7} = 53.4070751110265$$
$$x_{8} = 47.1238898038469$$
$$x_{9} = -97.3893722612836$$
$$x_{10} = 62.8318530717959$$
$$x_{11} = 87.9645943005142$$
$$x_{12} = 43.9822971502571$$
$$x_{13} = 37.6991118430775$$
$$x_{14} = -21.9911485751286$$
$$x_{15} = 3.14159265358979$$
$$x_{16} = 69.1150383789755$$
$$x_{17} = 65.9734457253857$$
$$x_{18} = -50.2654824574367$$
$$x_{19} = -94.2477796076938$$
$$x_{20} = -75.398223686155$$
$$x_{21} = -53.4070751110265$$
$$x_{22} = 12.5663706143592$$
$$x_{23} = -9.42477796076938$$
$$x_{24} = -34.5575191894877$$
$$x_{25} = 21.9911485751286$$
$$x_{26} = 446.106156809751$$
$$x_{27} = -47.1238898038469$$
$$x_{28} = -43.9822971502571$$
$$x_{29} = 28.2743338823081$$
$$x_{30} = -31.4159265358979$$
$$x_{31} = -3.14159265358979$$
$$x_{32} = -6.28318530717959$$
$$x_{33} = -25.1327412287183$$
$$x_{34} = -62.8318530717959$$
$$x_{35} = 31.4159265358979$$
$$x_{36} = -65.9734457253857$$
$$x_{37} = 72.2566310325652$$
$$x_{38} = -59.6902604182061$$
$$x_{39} = 94.2477796076938$$
$$x_{40} = 81.6814089933346$$
$$x_{41} = -91.106186954104$$
$$x_{42} = -100.530964914873$$
$$x_{43} = 59.6902604182061$$
$$x_{44} = -40.8407044966673$$
$$x_{45} = 91.106186954104$$
$$x_{46} = 78.5398163397448$$
$$x_{47} = -12.5663706143592$$
$$x_{48} = 56.5486677646163$$
$$x_{49} = 84.8230016469244$$
$$x_{50} = 100.530964914873$$
$$x_{51} = -69.1150383789755$$
$$x_{52} = 9.42477796076938$$
$$x_{53} = -84.8230016469244$$
$$x_{54} = -78.5398163397448$$
$$x_{55} = -87.9645943005142$$
$$x_{56} = -81.6814089933346$$
$$x_{57} = 15.707963267949$$
$$x_{58} = -28.2743338823081$$
$$x_{59} = -15.707963267949$$
$$x_{60} = -37.6991118430775$$
$$x_{61} = 179.070781254618$$
$$x_{62} = 18.8495559215388$$
$$x_{63} = 25.1327412287183$$
$$x_{64} = 50.2654824574367$$
$$x_{65} = -72.2566310325652$$
$$x_{66} = 75.398223686155$$
$$x_{67} = 6.28318530717959$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x \right)}}{x^{2} + x} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \pi$$
Solución numérica
$$x_{1} = 40.8407044966673$$
$$x_{2} = -135.088484104361$$
$$x_{3} = -18.8495559215388$$
$$x_{4} = -56.5486677646163$$
$$x_{5} = 97.3893722612836$$
$$x_{6} = 34.5575191894877$$
$$x_{7} = 53.4070751110265$$
$$x_{8} = 47.1238898038469$$
$$x_{9} = -97.3893722612836$$
$$x_{10} = 62.8318530717959$$
$$x_{11} = 87.9645943005142$$
$$x_{12} = 43.9822971502571$$
$$x_{13} = 37.6991118430775$$
$$x_{14} = -21.9911485751286$$
$$x_{15} = 3.14159265358979$$
$$x_{16} = 69.1150383789755$$
$$x_{17} = 65.9734457253857$$
$$x_{18} = -50.2654824574367$$
$$x_{19} = -94.2477796076938$$
$$x_{20} = -75.398223686155$$
$$x_{21} = -53.4070751110265$$
$$x_{22} = 12.5663706143592$$
$$x_{23} = -9.42477796076938$$
$$x_{24} = -34.5575191894877$$
$$x_{25} = 21.9911485751286$$
$$x_{26} = 446.106156809751$$
$$x_{27} = -47.1238898038469$$
$$x_{28} = -43.9822971502571$$
$$x_{29} = 28.2743338823081$$
$$x_{30} = -31.4159265358979$$
$$x_{31} = -3.14159265358979$$
$$x_{32} = -6.28318530717959$$
$$x_{33} = -25.1327412287183$$
$$x_{34} = -62.8318530717959$$
$$x_{35} = 31.4159265358979$$
$$x_{36} = -65.9734457253857$$
$$x_{37} = 72.2566310325652$$
$$x_{38} = -59.6902604182061$$
$$x_{39} = 94.2477796076938$$
$$x_{40} = 81.6814089933346$$
$$x_{41} = -91.106186954104$$
$$x_{42} = -100.530964914873$$
$$x_{43} = 59.6902604182061$$
$$x_{44} = -40.8407044966673$$
$$x_{45} = 91.106186954104$$
$$x_{46} = 78.5398163397448$$
$$x_{47} = -12.5663706143592$$
$$x_{48} = 56.5486677646163$$
$$x_{49} = 84.8230016469244$$
$$x_{50} = 100.530964914873$$
$$x_{51} = -69.1150383789755$$
$$x_{52} = 9.42477796076938$$
$$x_{53} = -84.8230016469244$$
$$x_{54} = -78.5398163397448$$
$$x_{55} = -87.9645943005142$$
$$x_{56} = -81.6814089933346$$
$$x_{57} = 15.707963267949$$
$$x_{58} = -28.2743338823081$$
$$x_{59} = -15.707963267949$$
$$x_{60} = -37.6991118430775$$
$$x_{61} = 179.070781254618$$
$$x_{62} = 18.8495559215388$$
$$x_{63} = 25.1327412287183$$
$$x_{64} = 50.2654824574367$$
$$x_{65} = -72.2566310325652$$
$$x_{66} = 75.398223686155$$
$$x_{67} = 6.28318530717959$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{\left(- 2 x - 1\right) \sin{\left(x \right)}}{\left(x^{2} + x\right)^{2}} + \frac{\cos{\left(x \right)}}{x^{2} + x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -45.5086813437597$$
$$x_{2} = -4.21030687099956$$
$$x_{3} = 26.6299264227573$$
$$x_{4} = 92.6555164857957$$
$$x_{5} = 83.2283222304467$$
$$x_{6} = 39.219589713028$$
$$x_{7} = 32.9269488002951$$
$$x_{8} = 70.6577341444619$$
$$x_{9} = -54.9411479381075$$
$$x_{10} = 86.3707785763499$$
$$x_{11} = -7.57725556612152$$
$$x_{12} = 117.792818621253$$
$$x_{13} = -10.8034064905843$$
$$x_{14} = 67.5148435894255$$
$$x_{15} = 95.797809455818$$
$$x_{16} = 7.61134921493137$$
$$x_{17} = -86.3705105117496$$
$$x_{18} = -29.7769031039576$$
$$x_{19} = 20.3245505305776$$
$$x_{20} = -20.3197194675214$$
$$x_{21} = -80.0854867532991$$
$$x_{22} = 48.6540159233106$$
$$x_{23} = -36.0721397966768$$
$$x_{24} = -32.9251056906799$$
$$x_{25} = 64.3718273069464$$
$$x_{26} = 76.9431992968741$$
$$x_{27} = 17.1659377579221$$
$$x_{28} = -95.7975915481518$$
$$x_{29} = 36.0736755951908$$
$$x_{30} = -17.1591707365239$$
$$x_{31} = -1110.55120132688$$
$$x_{32} = -67.5144049183161$$
$$x_{33} = -39.2182902732601$$
$$x_{34} = -51.7973064208209$$
$$x_{35} = 58.0853366248753$$
$$x_{36} = 10.8203712765393$$
$$x_{37} = -73.8001477840841$$
$$x_{38} = -13.989779179916$$
$$x_{39} = -23.4750719861659$$
$$x_{40} = 13.9999390615957$$
$$x_{41} = 42.3648700479129$$
$$x_{42} = -42.3637562976215$$
$$x_{43} = 98.9400581721213$$
$$x_{44} = -64.3713447630826$$
$$x_{45} = 51.7980515774978$$
$$x_{46} = -58.0847440097565$$
$$x_{47} = -48.653171390021$$
$$x_{48} = -26.6271098269614$$
$$x_{49} = 23.478694106118$$
$$x_{50} = 61.2286660937368$$
$$x_{51} = -218.331508225881$$
$$x_{52} = 80.0857985366819$$
$$x_{53} = -61.2281327485587$$
$$x_{54} = -274.882068187917$$
$$x_{55} = 54.9418102849687$$
$$x_{56} = 4.31483662620088$$
$$x_{57} = 73.8005149257302$$
$$x_{58} = -70.6573336227392$$
$$x_{59} = -76.9428615292039$$
$$x_{60} = -89.5129250514858$$
$$x_{61} = -98.9398538844255$$
$$x_{62} = 45.5096465557431$$
$$x_{63} = 152.354159938248$$
$$x_{64} = 29.7791560404789$$
$$x_{65} = 89.5131746276428$$
$$x_{66} = -117.792674488859$$
$$x_{67} = -92.6552835490818$$
$$x_{68} = -83.2280335439062$$
Signos de extremos en los puntos:
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:
$$x_{1} = -45.5086813437597$$
$$x_{2} = 92.6555164857957$$
$$x_{3} = 86.3707785763499$$
$$x_{4} = -7.57725556612152$$
$$x_{5} = 117.792818621253$$
$$x_{6} = 67.5148435894255$$
$$x_{7} = -20.3197194675214$$
$$x_{8} = 48.6540159233106$$
$$x_{9} = -32.9251056906799$$
$$x_{10} = 17.1659377579221$$
$$x_{11} = -95.7975915481518$$
$$x_{12} = 36.0736755951908$$
$$x_{13} = -39.2182902732601$$
$$x_{14} = -51.7973064208209$$
$$x_{15} = 10.8203712765393$$
$$x_{16} = -13.989779179916$$
$$x_{17} = 42.3648700479129$$
$$x_{18} = 98.9400581721213$$
$$x_{19} = -64.3713447630826$$
$$x_{20} = -58.0847440097565$$
$$x_{21} = -26.6271098269614$$
$$x_{22} = 23.478694106118$$
$$x_{23} = 61.2286660937368$$
$$x_{24} = 80.0857985366819$$
$$x_{25} = 54.9418102849687$$
$$x_{26} = 4.31483662620088$$
$$x_{27} = 73.8005149257302$$
$$x_{28} = -70.6573336227392$$
$$x_{29} = -76.9428615292039$$
$$x_{30} = -89.5129250514858$$
$$x_{31} = 29.7791560404789$$
$$x_{32} = -83.2280335439062$$
Puntos máximos de la función:
$$x_{32} = -4.21030687099956$$
$$x_{32} = 26.6299264227573$$
$$x_{32} = 83.2283222304467$$
$$x_{32} = 39.219589713028$$
$$x_{32} = 32.9269488002951$$
$$x_{32} = 70.6577341444619$$
$$x_{32} = -54.9411479381075$$
$$x_{32} = -10.8034064905843$$
$$x_{32} = 95.797809455818$$
$$x_{32} = 7.61134921493137$$
$$x_{32} = -86.3705105117496$$
$$x_{32} = -29.7769031039576$$
$$x_{32} = 20.3245505305776$$
$$x_{32} = -80.0854867532991$$
$$x_{32} = -36.0721397966768$$
$$x_{32} = 64.3718273069464$$
$$x_{32} = 76.9431992968741$$
$$x_{32} = -17.1591707365239$$
$$x_{32} = -1110.55120132688$$
$$x_{32} = -67.5144049183161$$
$$x_{32} = 58.0853366248753$$
$$x_{32} = -73.8001477840841$$
$$x_{32} = -23.4750719861659$$
$$x_{32} = 13.9999390615957$$
$$x_{32} = -42.3637562976215$$
$$x_{32} = 51.7980515774978$$
$$x_{32} = -48.653171390021$$
$$x_{32} = -218.331508225881$$
$$x_{32} = -61.2281327485587$$
$$x_{32} = -274.882068187917$$
$$x_{32} = -98.9398538844255$$
$$x_{32} = 45.5096465557431$$
$$x_{32} = 152.354159938248$$
$$x_{32} = 89.5131746276428$$
$$x_{32} = -117.792674488859$$
$$x_{32} = -92.6552835490818$$
Decrece en los intervalos
$$\left[117.792818621253, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.7975915481518\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{\left(- 2 x - 1\right) \sin{\left(x \right)}}{\left(x^{2} + x\right)^{2}} + \frac{\cos{\left(x \right)}}{x^{2} + x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -45.5086813437597$$
$$x_{2} = -4.21030687099956$$
$$x_{3} = 26.6299264227573$$
$$x_{4} = 92.6555164857957$$
$$x_{5} = 83.2283222304467$$
$$x_{6} = 39.219589713028$$
$$x_{7} = 32.9269488002951$$
$$x_{8} = 70.6577341444619$$
$$x_{9} = -54.9411479381075$$
$$x_{10} = 86.3707785763499$$
$$x_{11} = -7.57725556612152$$
$$x_{12} = 117.792818621253$$
$$x_{13} = -10.8034064905843$$
$$x_{14} = 67.5148435894255$$
$$x_{15} = 95.797809455818$$
$$x_{16} = 7.61134921493137$$
$$x_{17} = -86.3705105117496$$
$$x_{18} = -29.7769031039576$$
$$x_{19} = 20.3245505305776$$
$$x_{20} = -20.3197194675214$$
$$x_{21} = -80.0854867532991$$
$$x_{22} = 48.6540159233106$$
$$x_{23} = -36.0721397966768$$
$$x_{24} = -32.9251056906799$$
$$x_{25} = 64.3718273069464$$
$$x_{26} = 76.9431992968741$$
$$x_{27} = 17.1659377579221$$
$$x_{28} = -95.7975915481518$$
$$x_{29} = 36.0736755951908$$
$$x_{30} = -17.1591707365239$$
$$x_{31} = -1110.55120132688$$
$$x_{32} = -67.5144049183161$$
$$x_{33} = -39.2182902732601$$
$$x_{34} = -51.7973064208209$$
$$x_{35} = 58.0853366248753$$
$$x_{36} = 10.8203712765393$$
$$x_{37} = -73.8001477840841$$
$$x_{38} = -13.989779179916$$
$$x_{39} = -23.4750719861659$$
$$x_{40} = 13.9999390615957$$
$$x_{41} = 42.3648700479129$$
$$x_{42} = -42.3637562976215$$
$$x_{43} = 98.9400581721213$$
$$x_{44} = -64.3713447630826$$
$$x_{45} = 51.7980515774978$$
$$x_{46} = -58.0847440097565$$
$$x_{47} = -48.653171390021$$
$$x_{48} = -26.6271098269614$$
$$x_{49} = 23.478694106118$$
$$x_{50} = 61.2286660937368$$
$$x_{51} = -218.331508225881$$
$$x_{52} = 80.0857985366819$$
$$x_{53} = -61.2281327485587$$
$$x_{54} = -274.882068187917$$
$$x_{55} = 54.9418102849687$$
$$x_{56} = 4.31483662620088$$
$$x_{57} = 73.8005149257302$$
$$x_{58} = -70.6573336227392$$
$$x_{59} = -76.9428615292039$$
$$x_{60} = -89.5129250514858$$
$$x_{61} = -98.9398538844255$$
$$x_{62} = 45.5096465557431$$
$$x_{63} = 152.354159938248$$
$$x_{64} = 29.7791560404789$$
$$x_{65} = 89.5131746276428$$
$$x_{66} = -117.792674488859$$
$$x_{67} = -92.6552835490818$$
$$x_{68} = -83.2280335439062$$
Signos de extremos en los puntos:
(-45.50868134375969, -0.000493210791560672)
(-4.210306870999558, 0.0648533581866726)
(26.62992642275734, 0.00135541599741951)
(92.65551648579569, -0.000115211354183637)
(83.22832223044672, 0.000142608968015911)
(39.21958971302798, 0.000633153891451214)
(32.9269488002951, 0.000893567556392469)
(70.65773414446191, 0.000197426629471884)
(-54.94114793810752, 0.000337201247107331)
(86.3707785763499, -0.000132480477510489)
(-7.57725556612152, -0.0193018141186247)
(117.79281862125308, -7.14543928505629e-5)
(-10.803406490584345, 0.00926815978495185)
(67.51484358942552, -0.000216086861978215)
(95.79780945581795, 0.000107816502998618)
(7.611349214931373, 0.0148100372758499)
(-86.37051051174957, 0.000135584131658075)
(-29.77690310395756, 0.0011642998076989)
(20.32455053057763, 0.00229669400663664)
(-20.319719467521367, -0.00253442074006521)
(-80.08548675329911, 0.000157838260692457)
(48.654015923310595, -0.000413587736713029)
(-36.07213979667678, 0.000789187533574841)
(-32.92510569067994, -0.000949544928695192)
(64.3718273069464, 0.000237523844973448)
(76.94319929687411, 0.000166688923324479)
(17.165937757922052, -0.00318643265659244)
(-95.79759154815176, -0.000110091167941496)
(36.073675595190814, -0.000746612640870773)
(-17.15917073652391, 0.00358073081666456)
(-1110.5512013268826, 8.11546409922528e-7)
(-67.51440491831613, 0.000222584286098187)
(-39.2182902732601, -0.000666286882142444)
(-51.79730642082091, -0.000379771389388018)
(58.0853366248753, 0.000291206334797043)
(10.820371276539255, -0.00769886605705645)
(-73.80014778408407, 0.000186058232912999)
(-13.989779179916034, -0.00544318470484132)
(-23.47507198616592, 0.00188821359913657)
(13.999939061595747, 0.00471717789310325)
(42.36487004791293, -0.000543730543381711)
(-42.363756297621535, 0.000570020381249577)
(98.94005817212127, -0.000101111467629888)
(-64.3713447630826, -0.000245020070327294)
(51.79805157749776, 0.000365385477105757)
(-58.084744009756506, -0.000301408879392996)
(-48.653171390021036, 0.000430945828935977)
(-26.6271098269614, -0.00146119004117301)
(23.478694106118, -0.00173392813728352)
(61.22866609373683, -0.000262317249117401)
(-218.331508225881, 2.10737870238563e-5)
(80.08579853668186, -0.000153945135953131)
(-61.228132748558664, 0.000271027998585765)
(-274.8820681879171, 1.3282458096703e-5)
(54.94181028496868, -0.000325145747236153)
(4.314836626200884, -0.0402051800541067)
(73.8005149257302, -0.00018108343588889)
(-70.6573336227392, -0.00020309512269768)
(-76.9428615292039, -0.000171078764562527)
(-89.51292505148578, -0.000126182159056221)
(-98.9398538844255, 0.000103176232037189)
(45.50964655574305, 0.000472001603964367)
(152.35415993824822, 4.27969627227201e-5)
(29.779156040478902, -0.00108864195536588)
(89.51317462764281, 0.000123394005098754)
(-117.79267448885864, 7.26780029626325e-5)
(-92.65528354908184, 0.000117725365451097)
(-83.22803354390616, -0.000146077583560614)
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:
$$x_{1} = -45.5086813437597$$
$$x_{2} = 92.6555164857957$$
$$x_{3} = 86.3707785763499$$
$$x_{4} = -7.57725556612152$$
$$x_{5} = 117.792818621253$$
$$x_{6} = 67.5148435894255$$
$$x_{7} = -20.3197194675214$$
$$x_{8} = 48.6540159233106$$
$$x_{9} = -32.9251056906799$$
$$x_{10} = 17.1659377579221$$
$$x_{11} = -95.7975915481518$$
$$x_{12} = 36.0736755951908$$
$$x_{13} = -39.2182902732601$$
$$x_{14} = -51.7973064208209$$
$$x_{15} = 10.8203712765393$$
$$x_{16} = -13.989779179916$$
$$x_{17} = 42.3648700479129$$
$$x_{18} = 98.9400581721213$$
$$x_{19} = -64.3713447630826$$
$$x_{20} = -58.0847440097565$$
$$x_{21} = -26.6271098269614$$
$$x_{22} = 23.478694106118$$
$$x_{23} = 61.2286660937368$$
$$x_{24} = 80.0857985366819$$
$$x_{25} = 54.9418102849687$$
$$x_{26} = 4.31483662620088$$
$$x_{27} = 73.8005149257302$$
$$x_{28} = -70.6573336227392$$
$$x_{29} = -76.9428615292039$$
$$x_{30} = -89.5129250514858$$
$$x_{31} = 29.7791560404789$$
$$x_{32} = -83.2280335439062$$
Puntos máximos de la función:
$$x_{32} = -4.21030687099956$$
$$x_{32} = 26.6299264227573$$
$$x_{32} = 83.2283222304467$$
$$x_{32} = 39.219589713028$$
$$x_{32} = 32.9269488002951$$
$$x_{32} = 70.6577341444619$$
$$x_{32} = -54.9411479381075$$
$$x_{32} = -10.8034064905843$$
$$x_{32} = 95.797809455818$$
$$x_{32} = 7.61134921493137$$
$$x_{32} = -86.3705105117496$$
$$x_{32} = -29.7769031039576$$
$$x_{32} = 20.3245505305776$$
$$x_{32} = -80.0854867532991$$
$$x_{32} = -36.0721397966768$$
$$x_{32} = 64.3718273069464$$
$$x_{32} = 76.9431992968741$$
$$x_{32} = -17.1591707365239$$
$$x_{32} = -1110.55120132688$$
$$x_{32} = -67.5144049183161$$
$$x_{32} = 58.0853366248753$$
$$x_{32} = -73.8001477840841$$
$$x_{32} = -23.4750719861659$$
$$x_{32} = 13.9999390615957$$
$$x_{32} = -42.3637562976215$$
$$x_{32} = 51.7980515774978$$
$$x_{32} = -48.653171390021$$
$$x_{32} = -218.331508225881$$
$$x_{32} = -61.2281327485587$$
$$x_{32} = -274.882068187917$$
$$x_{32} = -98.9398538844255$$
$$x_{32} = 45.5096465557431$$
$$x_{32} = 152.354159938248$$
$$x_{32} = 89.5131746276428$$
$$x_{32} = -117.792674488859$$
$$x_{32} = -92.6552835490818$$
Decrece en los intervalos
$$\left[117.792818621253, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.7975915481518\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$- \frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \sin{\left(x \right)}}{x \left(x + 1\right)} + \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{x \left(x + 1\right)}}{x \left(x + 1\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -8.94579798748416$$
$$x_{2} = -62.7675990506214$$
$$x_{3} = 81.6327007098498$$
$$x_{4} = -72.2008337448729$$
$$x_{5} = -103.633769648051$$
$$x_{6} = -53.3313376817066$$
$$x_{7} = 72.2016020855235$$
$$x_{8} = 97.3484888232346$$
$$x_{9} = -81.6320998255227$$
$$x_{10} = 8.99983911516347$$
$$x_{11} = 43.8921493941254$$
$$x_{12} = 34.4429615778181$$
$$x_{13} = -65.9122820550677$$
$$x_{14} = -122.489321716127$$
$$x_{15} = 24.975508794385$$
$$x_{16} = -56.4771892378559$$
$$x_{17} = 1529.95300869197$$
$$x_{18} = 62.7686161257064$$
$$x_{19} = -15.4391522275548$$
$$x_{20} = 37.5940426051172$$
$$x_{21} = -47.0379021201791$$
$$x_{22} = -97.3480664230007$$
$$x_{23} = -43.8900654657285$$
$$x_{24} = -69.0566811383483$$
$$x_{25} = 91.0624961736649$$
$$x_{26} = -40.7412480268036$$
$$x_{27} = 65.9132042453391$$
$$x_{28} = 100.491354007135$$
$$x_{29} = -31.2858722611977$$
$$x_{30} = 47.0397156288495$$
$$x_{31} = -24.9690224794265$$
$$x_{32} = 75.3454764868482$$
$$x_{33} = -18.6283068532307$$
$$x_{34} = 50.1865379748233$$
$$x_{35} = -94.205088039205$$
$$x_{36} = -100.490957633034$$
$$x_{37} = -91.0620133932049$$
$$x_{38} = -75.3447710064488$$
$$x_{39} = 5.61676503532271$$
$$x_{40} = 15.4564076125645$$
$$x_{41} = -59.6225866802072$$
$$x_{42} = -84.7755321674264$$
$$x_{43} = 69.0575211405475$$
$$x_{44} = 94.2055391172032$$
$$x_{45} = -34.4395695544194$$
$$x_{46} = 78.4891690557357$$
$$x_{47} = 40.7436679077131$$
$$x_{48} = -12.2230251941864$$
$$x_{49} = 84.7760892748272$$
$$x_{50} = 31.289987578162$$
$$x_{51} = -273.303898134652$$
$$x_{52} = -37.5911981316645$$
$$x_{53} = 21.8115446559168$$
$$x_{54} = -5.45589719485634$$
$$x_{55} = 12.2509924817039$$
$$x_{56} = -87.9188320868598$$
$$x_{57} = -50.184945380919$$
$$x_{58} = -28.1293885794892$$
$$x_{59} = -21.8030094559639$$
$$x_{60} = 28.1344876262722$$
$$x_{61} = -78.4885190197697$$
$$x_{62} = 53.3327474617875$$
$$x_{63} = 87.9193500355846$$
$$x_{64} = 18.6400592617187$$
$$x_{65} = 59.6237140974563$$
$$x_{66} = 56.478446009717$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -1$$
$$x_{2} = 0$$
$$\lim_{x \to -1^-}\left(- \frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \sin{\left(x \right)}}{x \left(x + 1\right)} + \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{x \left(x + 1\right)}}{x \left(x + 1\right)}\right) = -\infty$$
$$\lim_{x \to -1^+}\left(- \frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \sin{\left(x \right)}}{x \left(x + 1\right)} + \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{x \left(x + 1\right)}}{x \left(x + 1\right)}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = -1$$
- es el punto de flexión
$$\lim_{x \to 0^-}\left(- \frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \sin{\left(x \right)}}{x \left(x + 1\right)} + \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{x \left(x + 1\right)}}{x \left(x + 1\right)}\right) = \frac{5}{3}$$
$$\lim_{x \to 0^+}\left(- \frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \sin{\left(x \right)}}{x \left(x + 1\right)} + \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{x \left(x + 1\right)}}{x \left(x + 1\right)}\right) = \frac{5}{3}$$
- los límites son iguales, es decir omitimos el punto correspondiente
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[1529.95300869197, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -273.303898134652\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$- \frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \sin{\left(x \right)}}{x \left(x + 1\right)} + \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{x \left(x + 1\right)}}{x \left(x + 1\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -8.94579798748416$$
$$x_{2} = -62.7675990506214$$
$$x_{3} = 81.6327007098498$$
$$x_{4} = -72.2008337448729$$
$$x_{5} = -103.633769648051$$
$$x_{6} = -53.3313376817066$$
$$x_{7} = 72.2016020855235$$
$$x_{8} = 97.3484888232346$$
$$x_{9} = -81.6320998255227$$
$$x_{10} = 8.99983911516347$$
$$x_{11} = 43.8921493941254$$
$$x_{12} = 34.4429615778181$$
$$x_{13} = -65.9122820550677$$
$$x_{14} = -122.489321716127$$
$$x_{15} = 24.975508794385$$
$$x_{16} = -56.4771892378559$$
$$x_{17} = 1529.95300869197$$
$$x_{18} = 62.7686161257064$$
$$x_{19} = -15.4391522275548$$
$$x_{20} = 37.5940426051172$$
$$x_{21} = -47.0379021201791$$
$$x_{22} = -97.3480664230007$$
$$x_{23} = -43.8900654657285$$
$$x_{24} = -69.0566811383483$$
$$x_{25} = 91.0624961736649$$
$$x_{26} = -40.7412480268036$$
$$x_{27} = 65.9132042453391$$
$$x_{28} = 100.491354007135$$
$$x_{29} = -31.2858722611977$$
$$x_{30} = 47.0397156288495$$
$$x_{31} = -24.9690224794265$$
$$x_{32} = 75.3454764868482$$
$$x_{33} = -18.6283068532307$$
$$x_{34} = 50.1865379748233$$
$$x_{35} = -94.205088039205$$
$$x_{36} = -100.490957633034$$
$$x_{37} = -91.0620133932049$$
$$x_{38} = -75.3447710064488$$
$$x_{39} = 5.61676503532271$$
$$x_{40} = 15.4564076125645$$
$$x_{41} = -59.6225866802072$$
$$x_{42} = -84.7755321674264$$
$$x_{43} = 69.0575211405475$$
$$x_{44} = 94.2055391172032$$
$$x_{45} = -34.4395695544194$$
$$x_{46} = 78.4891690557357$$
$$x_{47} = 40.7436679077131$$
$$x_{48} = -12.2230251941864$$
$$x_{49} = 84.7760892748272$$
$$x_{50} = 31.289987578162$$
$$x_{51} = -273.303898134652$$
$$x_{52} = -37.5911981316645$$
$$x_{53} = 21.8115446559168$$
$$x_{54} = -5.45589719485634$$
$$x_{55} = 12.2509924817039$$
$$x_{56} = -87.9188320868598$$
$$x_{57} = -50.184945380919$$
$$x_{58} = -28.1293885794892$$
$$x_{59} = -21.8030094559639$$
$$x_{60} = 28.1344876262722$$
$$x_{61} = -78.4885190197697$$
$$x_{62} = 53.3327474617875$$
$$x_{63} = 87.9193500355846$$
$$x_{64} = 18.6400592617187$$
$$x_{65} = 59.6237140974563$$
$$x_{66} = 56.478446009717$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -1$$
$$x_{2} = 0$$
$$\lim_{x \to -1^-}\left(- \frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \sin{\left(x \right)}}{x \left(x + 1\right)} + \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{x \left(x + 1\right)}}{x \left(x + 1\right)}\right) = -\infty$$
$$\lim_{x \to -1^+}\left(- \frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \sin{\left(x \right)}}{x \left(x + 1\right)} + \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{x \left(x + 1\right)}}{x \left(x + 1\right)}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = -1$$
- es el punto de flexión
$$\lim_{x \to 0^-}\left(- \frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \sin{\left(x \right)}}{x \left(x + 1\right)} + \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{x \left(x + 1\right)}}{x \left(x + 1\right)}\right) = \frac{5}{3}$$
$$\lim_{x \to 0^+}\left(- \frac{\sin{\left(x \right)} + \frac{2 \left(1 - \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \sin{\left(x \right)}}{x \left(x + 1\right)} + \frac{2 \left(2 x + 1\right) \cos{\left(x \right)}}{x \left(x + 1\right)}}{x \left(x + 1\right)}\right) = \frac{5}{3}$$
- los límites son iguales, es decir omitimos el punto correspondiente
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[1529.95300869197, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -273.303898134652\right]$$
Asíntotas verticales
Hay:
$$x_{1} = -1$$
$$x_{2} = 0$$
$$x_{1} = -1$$
$$x_{2} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x^{2} + x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x^{2} + x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x^{2} + x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x^{2} + x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)/(x^2 + x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(x^{2} + x\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(x^{2} + x\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(x^{2} + x\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(x^{2} + x\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
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:
$$\frac{\sin{\left(x \right)}}{x^{2} + x} = - \frac{\sin{\left(x \right)}}{x^{2} - x}$$
- No
$$\frac{\sin{\left(x \right)}}{x^{2} + x} = \frac{\sin{\left(x \right)}}{x^{2} - x}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(x \right)}}{x^{2} + x} = - \frac{\sin{\left(x \right)}}{x^{2} - x}$$
- No
$$\frac{\sin{\left(x \right)}}{x^{2} + x} = \frac{\sin{\left(x \right)}}{x^{2} - x}$$
- No
es decir, función
no es
par ni impar