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- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
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- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x+27/x^3
-
(x^2-8)/(x-3)
-
x-2+4/(x-2)
-
x^2-9*x+14
- Integral de d{x}:
- (x^2+x)*sin(x)
-
Expresiones idénticas
- (x^ dos +x)*sin(x)
- (x al cuadrado más x) multiplicar por seno de (x)
- (x en el grado dos más x) multiplicar por seno de (x)
- (x2+x)*sin(x)
- x2+x*sinx
- (x²+x)*sin(x)
- (x en el grado 2+x)*sin(x)
- (x^2+x)sin(x)
- (x2+x)sin(x)
- x2+xsinx
- x^2+xsinx
-
Expresiones semejantes
- (x^2-x)*sin(x)
- (x^2+x)*sinx
-
Expresiones con funciones
- Seno sin
- sin(x-pi/4)*(-2)
- sin(pi*x)/asin(x)
- sin(x)*e^((1/2)*cot(x)^(2))
- sin(697*t)+sin(1209*t)
- sin(3x-29)
Gráfico de la función y = (x^2+x)*sin(x)
Solución
Ha introducido
[src]
/ 2 \ f(x) = \x + x/*sin(x)
$$f{\left(x \right)} = \left(x^{2} + x\right) \sin{\left(x \right)}$$
f = (x^2 + x)*sin(x)
Gráfico de la función
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:
$$\left(x^{2} + x\right) \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = -1$$
$$x_{2} = 0$$
$$x_{3} = \pi$$
Solución numérica
$$x_{1} = 69.1150383789755$$
$$x_{2} = 65.9734457253857$$
$$x_{3} = -91.106186954104$$
$$x_{4} = -103.672557568463$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = -21.9911485751286$$
$$x_{7} = 12.5663706143592$$
$$x_{8} = 21.9911485751286$$
$$x_{9} = -69.1150383789755$$
$$x_{10} = -100.530964914873$$
$$x_{11} = 3.14159265358979$$
$$x_{12} = -1$$
$$x_{13} = -3.14159265358979$$
$$x_{14} = -25.1327412287183$$
$$x_{15} = -116.238928182822$$
$$x_{16} = -53.4070751110265$$
$$x_{17} = -15.707963267949$$
$$x_{18} = -72.2566310325652$$
$$x_{19} = 84.8230016469244$$
$$x_{20} = -81.6814089933346$$
$$x_{21} = -94.2477796076938$$
$$x_{22} = 18.8495559215388$$
$$x_{23} = -65.9734457253857$$
$$x_{24} = 94.2477796076938$$
$$x_{25} = 9.42477796076938$$
$$x_{26} = -40.8407044966673$$
$$x_{27} = 34.5575191894877$$
$$x_{28} = 0$$
$$x_{29} = 97.3893722612836$$
$$x_{30} = 53.4070751110265$$
$$x_{31} = 147.65485471872$$
$$x_{32} = -62.8318530717959$$
$$x_{33} = 59.6902604182061$$
$$x_{34} = -28.2743338823081$$
$$x_{35} = -56.5486677646163$$
$$x_{36} = 91.106186954104$$
$$x_{37} = 15.707963267949$$
$$x_{38} = -18.8495559215388$$
$$x_{39} = 6.28318530717959$$
$$x_{40} = 56.5486677646163$$
$$x_{41} = 87.9645943005142$$
$$x_{42} = 31.4159265358979$$
$$x_{43} = 25.1327412287183$$
$$x_{44} = 43.9822971502571$$
$$x_{45} = -47.1238898038469$$
$$x_{46} = 72.2566310325652$$
$$x_{47} = -34.5575191894877$$
$$x_{48} = -97.3893722612836$$
$$x_{49} = -50.2654824574367$$
$$x_{50} = 100.530964914873$$
$$x_{51} = 81.6814089933346$$
$$x_{52} = -75.398223686155$$
$$x_{53} = 40.8407044966673$$
$$x_{54} = -9.42477796076938$$
$$x_{55} = 78.5398163397448$$
$$x_{56} = -87.9645943005142$$
$$x_{57} = 37.6991118430775$$
$$x_{58} = -78.5398163397448$$
$$x_{59} = -6.28318530717959$$
$$x_{60} = 50.2654824574367$$
$$x_{61} = -37.6991118430775$$
$$x_{62} = -43.9822971502571$$
$$x_{63} = 47.1238898038469$$
$$x_{64} = 28.2743338823081$$
$$x_{65} = 62.8318530717959$$
$$x_{66} = -31.4159265358979$$
$$x_{67} = -12.5663706143592$$
$$x_{68} = 75.398223686155$$
$$x_{69} = -84.8230016469244$$
o sea hay que resolver la ecuación:
$$\left(x^{2} + x\right) \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = -1$$
$$x_{2} = 0$$
$$x_{3} = \pi$$
Solución numérica
$$x_{1} = 69.1150383789755$$
$$x_{2} = 65.9734457253857$$
$$x_{3} = -91.106186954104$$
$$x_{4} = -103.672557568463$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = -21.9911485751286$$
$$x_{7} = 12.5663706143592$$
$$x_{8} = 21.9911485751286$$
$$x_{9} = -69.1150383789755$$
$$x_{10} = -100.530964914873$$
$$x_{11} = 3.14159265358979$$
$$x_{12} = -1$$
$$x_{13} = -3.14159265358979$$
$$x_{14} = -25.1327412287183$$
$$x_{15} = -116.238928182822$$
$$x_{16} = -53.4070751110265$$
$$x_{17} = -15.707963267949$$
$$x_{18} = -72.2566310325652$$
$$x_{19} = 84.8230016469244$$
$$x_{20} = -81.6814089933346$$
$$x_{21} = -94.2477796076938$$
$$x_{22} = 18.8495559215388$$
$$x_{23} = -65.9734457253857$$
$$x_{24} = 94.2477796076938$$
$$x_{25} = 9.42477796076938$$
$$x_{26} = -40.8407044966673$$
$$x_{27} = 34.5575191894877$$
$$x_{28} = 0$$
$$x_{29} = 97.3893722612836$$
$$x_{30} = 53.4070751110265$$
$$x_{31} = 147.65485471872$$
$$x_{32} = -62.8318530717959$$
$$x_{33} = 59.6902604182061$$
$$x_{34} = -28.2743338823081$$
$$x_{35} = -56.5486677646163$$
$$x_{36} = 91.106186954104$$
$$x_{37} = 15.707963267949$$
$$x_{38} = -18.8495559215388$$
$$x_{39} = 6.28318530717959$$
$$x_{40} = 56.5486677646163$$
$$x_{41} = 87.9645943005142$$
$$x_{42} = 31.4159265358979$$
$$x_{43} = 25.1327412287183$$
$$x_{44} = 43.9822971502571$$
$$x_{45} = -47.1238898038469$$
$$x_{46} = 72.2566310325652$$
$$x_{47} = -34.5575191894877$$
$$x_{48} = -97.3893722612836$$
$$x_{49} = -50.2654824574367$$
$$x_{50} = 100.530964914873$$
$$x_{51} = 81.6814089933346$$
$$x_{52} = -75.398223686155$$
$$x_{53} = 40.8407044966673$$
$$x_{54} = -9.42477796076938$$
$$x_{55} = 78.5398163397448$$
$$x_{56} = -87.9645943005142$$
$$x_{57} = 37.6991118430775$$
$$x_{58} = -78.5398163397448$$
$$x_{59} = -6.28318530717959$$
$$x_{60} = 50.2654824574367$$
$$x_{61} = -37.6991118430775$$
$$x_{62} = -43.9822971502571$$
$$x_{63} = 47.1238898038469$$
$$x_{64} = 28.2743338823081$$
$$x_{65} = 62.8318530717959$$
$$x_{66} = -31.4159265358979$$
$$x_{67} = -12.5663706143592$$
$$x_{68} = 75.398223686155$$
$$x_{69} = -84.8230016469244$$
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
$$\left(2 x + 1\right) \sin{\left(x \right)} + \left(x^{2} + x\right) \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -48.7361295392441$$
$$x_{2} = -45.5974183148387$$
$$x_{3} = 20.5152887094394$$
$$x_{4} = -42.4591369509051$$
$$x_{5} = -92.6986728032932$$
$$x_{6} = -23.6481708154477$$
$$x_{7} = 58.1535518920027$$
$$x_{8} = -67.5740526651947$$
$$x_{9} = 14.2718933831386$$
$$x_{10} = 2.22109820882798$$
$$x_{11} = -26.779523413937$$
$$x_{12} = 64.4334422583637$$
$$x_{13} = 86.4168051847116$$
$$x_{14} = -11.1810751482291$$
$$x_{15} = 29.910819411552$$
$$x_{16} = 42.4580305922627$$
$$x_{17} = -0.649755168374399$$
$$x_{18} = 45.5964586594414$$
$$x_{19} = -55.0145455295839$$
$$x_{20} = 26.7767536014234$$
$$x_{21} = 51.8744502729041$$
$$x_{22} = -17.3966801527861$$
$$x_{23} = 33.046284696792$$
$$x_{24} = -95.8395510876656$$
$$x_{25} = 39.3200997911221$$
$$x_{26} = -83.2763629879876$$
$$x_{27} = 92.6984401916674$$
$$x_{28} = 55.0138858041882$$
$$x_{29} = 0$$
$$x_{30} = -2.41352739608161$$
$$x_{31} = -58.1541424076718$$
$$x_{32} = 98.9802708766234$$
$$x_{33} = 89.5575956718273$$
$$x_{34} = -29.9130423127956$$
$$x_{35} = 11.1656732464925$$
$$x_{36} = 76.9948235344069$$
$$x_{37} = 5.06026836518236$$
$$x_{38} = -89.5578448748316$$
$$x_{39} = 95.8393334645179$$
$$x_{40} = -36.1843149453611$$
$$x_{41} = -8.1119826164815$$
$$x_{42} = -33.0481076541221$$
$$x_{43} = -64.4339234092994$$
$$x_{44} = 70.7139131086823$$
$$x_{45} = -14.2814705578331$$
$$x_{46} = -80.1357227352106$$
$$x_{47} = -76.9951606189843$$
$$x_{48} = 23.6446256872296$$
$$x_{49} = -39.3213891805148$$
$$x_{50} = -70.7143126702811$$
$$x_{51} = -51.8751921133361$$
$$x_{52} = 67.5736151455506$$
$$x_{53} = -158.663073635328$$
$$x_{54} = 36.1827931655659$$
$$x_{55} = 83.276074800594$$
$$x_{56} = 61.2934134764651$$
$$x_{57} = 80.1354115339569$$
$$x_{58} = -20.519983804286$$
$$x_{59} = -86.4170728188749$$
$$x_{60} = -5.12485035716986$$
$$x_{61} = 8.08365076227732$$
$$x_{62} = 48.7352892631046$$
$$x_{63} = -98.9804749142348$$
$$x_{64} = 73.8543203258377$$
$$x_{65} = 17.3901768408315$$
$$x_{66} = -61.2939451205206$$
$$x_{67} = -73.854686660561$$
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:
$$x_{1} = -45.5974183148387$$
$$x_{2} = -26.779523413937$$
$$x_{3} = 86.4168051847116$$
$$x_{4} = 29.910819411552$$
$$x_{5} = 42.4580305922627$$
$$x_{6} = -95.8395510876656$$
$$x_{7} = -83.2763629879876$$
$$x_{8} = 92.6984401916674$$
$$x_{9} = 55.0138858041882$$
$$x_{10} = 0$$
$$x_{11} = -2.41352739608161$$
$$x_{12} = -58.1541424076718$$
$$x_{13} = 98.9802708766234$$
$$x_{14} = 11.1656732464925$$
$$x_{15} = 5.06026836518236$$
$$x_{16} = -89.5578448748316$$
$$x_{17} = -8.1119826164815$$
$$x_{18} = -33.0481076541221$$
$$x_{19} = -64.4339234092994$$
$$x_{20} = -14.2814705578331$$
$$x_{21} = -76.9951606189843$$
$$x_{22} = 23.6446256872296$$
$$x_{23} = -39.3213891805148$$
$$x_{24} = -70.7143126702811$$
$$x_{25} = -51.8751921133361$$
$$x_{26} = 67.5736151455506$$
$$x_{27} = -158.663073635328$$
$$x_{28} = 36.1827931655659$$
$$x_{29} = 61.2934134764651$$
$$x_{30} = 80.1354115339569$$
$$x_{31} = -20.519983804286$$
$$x_{32} = 48.7352892631046$$
$$x_{33} = 73.8543203258377$$
$$x_{34} = 17.3901768408315$$
Puntos máximos de la función:
$$x_{34} = -48.7361295392441$$
$$x_{34} = 20.5152887094394$$
$$x_{34} = -42.4591369509051$$
$$x_{34} = -92.6986728032932$$
$$x_{34} = -23.6481708154477$$
$$x_{34} = 58.1535518920027$$
$$x_{34} = -67.5740526651947$$
$$x_{34} = 14.2718933831386$$
$$x_{34} = 2.22109820882798$$
$$x_{34} = 64.4334422583637$$
$$x_{34} = -11.1810751482291$$
$$x_{34} = -0.649755168374399$$
$$x_{34} = 45.5964586594414$$
$$x_{34} = -55.0145455295839$$
$$x_{34} = 26.7767536014234$$
$$x_{34} = 51.8744502729041$$
$$x_{34} = -17.3966801527861$$
$$x_{34} = 33.046284696792$$
$$x_{34} = 39.3200997911221$$
$$x_{34} = 89.5575956718273$$
$$x_{34} = -29.9130423127956$$
$$x_{34} = 76.9948235344069$$
$$x_{34} = 95.8393334645179$$
$$x_{34} = -36.1843149453611$$
$$x_{34} = 70.7139131086823$$
$$x_{34} = -80.1357227352106$$
$$x_{34} = 83.276074800594$$
$$x_{34} = -86.4170728188749$$
$$x_{34} = -5.12485035716986$$
$$x_{34} = 8.08365076227732$$
$$x_{34} = -98.9804749142348$$
$$x_{34} = -61.2939451205206$$
$$x_{34} = -73.854686660561$$
Decrece en los intervalos
$$\left[98.9802708766234, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -158.663073635328\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\left(2 x + 1\right) \sin{\left(x \right)} + \left(x^{2} + x\right) \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -48.7361295392441$$
$$x_{2} = -45.5974183148387$$
$$x_{3} = 20.5152887094394$$
$$x_{4} = -42.4591369509051$$
$$x_{5} = -92.6986728032932$$
$$x_{6} = -23.6481708154477$$
$$x_{7} = 58.1535518920027$$
$$x_{8} = -67.5740526651947$$
$$x_{9} = 14.2718933831386$$
$$x_{10} = 2.22109820882798$$
$$x_{11} = -26.779523413937$$
$$x_{12} = 64.4334422583637$$
$$x_{13} = 86.4168051847116$$
$$x_{14} = -11.1810751482291$$
$$x_{15} = 29.910819411552$$
$$x_{16} = 42.4580305922627$$
$$x_{17} = -0.649755168374399$$
$$x_{18} = 45.5964586594414$$
$$x_{19} = -55.0145455295839$$
$$x_{20} = 26.7767536014234$$
$$x_{21} = 51.8744502729041$$
$$x_{22} = -17.3966801527861$$
$$x_{23} = 33.046284696792$$
$$x_{24} = -95.8395510876656$$
$$x_{25} = 39.3200997911221$$
$$x_{26} = -83.2763629879876$$
$$x_{27} = 92.6984401916674$$
$$x_{28} = 55.0138858041882$$
$$x_{29} = 0$$
$$x_{30} = -2.41352739608161$$
$$x_{31} = -58.1541424076718$$
$$x_{32} = 98.9802708766234$$
$$x_{33} = 89.5575956718273$$
$$x_{34} = -29.9130423127956$$
$$x_{35} = 11.1656732464925$$
$$x_{36} = 76.9948235344069$$
$$x_{37} = 5.06026836518236$$
$$x_{38} = -89.5578448748316$$
$$x_{39} = 95.8393334645179$$
$$x_{40} = -36.1843149453611$$
$$x_{41} = -8.1119826164815$$
$$x_{42} = -33.0481076541221$$
$$x_{43} = -64.4339234092994$$
$$x_{44} = 70.7139131086823$$
$$x_{45} = -14.2814705578331$$
$$x_{46} = -80.1357227352106$$
$$x_{47} = -76.9951606189843$$
$$x_{48} = 23.6446256872296$$
$$x_{49} = -39.3213891805148$$
$$x_{50} = -70.7143126702811$$
$$x_{51} = -51.8751921133361$$
$$x_{52} = 67.5736151455506$$
$$x_{53} = -158.663073635328$$
$$x_{54} = 36.1827931655659$$
$$x_{55} = 83.276074800594$$
$$x_{56} = 61.2934134764651$$
$$x_{57} = 80.1354115339569$$
$$x_{58} = -20.519983804286$$
$$x_{59} = -86.4170728188749$$
$$x_{60} = -5.12485035716986$$
$$x_{61} = 8.08365076227732$$
$$x_{62} = 48.7352892631046$$
$$x_{63} = -98.9804749142348$$
$$x_{64} = 73.8543203258377$$
$$x_{65} = 17.3901768408315$$
$$x_{66} = -61.2939451205206$$
$$x_{67} = -73.854686660561$$
Signos de extremos en los puntos:
(-48.73612953924414, 2324.47655388266)
(-45.59741831483875, -2031.52983921935)
(20.515288709439442, 439.404733492473)
(-42.45913695090512, 1758.3222926222)
(-92.6986728032932, 8498.34591348608)
(-23.648170815447703, 533.598022286833)
(58.153551892002696, 3437.99074695487)
(-67.57405266519466, 4496.67976267516)
(14.271893383138554, 215.983715663497)
(2.221098208827979, 5.69417506667367)
(-26.779523413937035, -688.371282181144)
(64.43344225836368, 4214.10322708716)
(86.41680518471165, -7552.28175128335)
(-11.181075148229146, 111.882406294904)
(29.91081941155198, -922.573866189311)
(42.45803059226273, -1843.14536818102)
(-0.6497551683743993, 0.137679962001981)
(45.59645865944136, 2122.63608586311)
(-55.0145455295839, 2969.58752323768)
(26.776753601423422, 741.778651182549)
(51.87445027290409, 2740.83504435784)
(-17.396680152786104, 283.266875092874)
(33.046284696792036, 1123.10809200774)
(-95.8395510876656, -9087.38060649368)
(39.32009979112207, 1583.39380980238)
(-83.27636298798761, -6849.67707188041)
(92.69844019166737, -8683.69988715932)
(55.01388580418817, -3079.54330011308)
(0, 0)
(-2.4135273960816113, -2.2701610308603)
(-58.154142407671834, -3321.75178998414)
(98.98027087662336, -9894.07484928877)
(89.55759567182727, 8108.12121609698)
(-29.913042312795604, 862.883394738015)
(11.16567324649254, -133.877521969079)
(76.99482353440692, 6003.19859003098)
(5.060268365182357, -28.8295870295239)
(-89.55784487483163, -7929.05042695635)
(95.83933346451788, 9279.01776479774)
(-36.18431494536108, 1271.12464275245)
(-8.111982616481502, -55.7827777203279)
(-33.04810765412208, -1057.13448970071)
(-64.43392340929937, -4085.29790712209)
(70.71391310868229, 5069.17250415237)
(-14.281470557833119, -187.707462567745)
(-80.1357227352106, 6339.59920242608)
(-76.99516061898431, -5849.2605375965)
(23.644625687229603, -580.722338398605)
(-39.32138918051478, -1504.85390043184)
(-70.71431267028109, -4927.80081872494)
(-51.875192113336084, -2637.16244624062)
(67.57361514555056, -4631.7682651326)
(-158.66307363532783, -25013.3080816123)
(36.18279316556589, -1343.38139313081)
(83.276074800594, 7016.18149232862)
(61.2934134764651, -3816.17738838667)
(80.13541153395687, -6499.82043876569)
(-20.519983804286046, -398.56337754808)
(-86.41707281887494, 7379.49414655867)
(-5.124850357169864, 19.3664389379329)
(8.083650762277319, 71.5009434734222)
(48.735289263104576, -2421.86597503638)
(-98.98047491423482, 9696.15450627079)
(73.85432032583775, -5526.31594543957)
(17.390176840831522, -317.825460652734)
(-61.29394512052059, 3693.65525030791)
(-73.854686660561, 5378.66107666168)
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:
$$x_{1} = -45.5974183148387$$
$$x_{2} = -26.779523413937$$
$$x_{3} = 86.4168051847116$$
$$x_{4} = 29.910819411552$$
$$x_{5} = 42.4580305922627$$
$$x_{6} = -95.8395510876656$$
$$x_{7} = -83.2763629879876$$
$$x_{8} = 92.6984401916674$$
$$x_{9} = 55.0138858041882$$
$$x_{10} = 0$$
$$x_{11} = -2.41352739608161$$
$$x_{12} = -58.1541424076718$$
$$x_{13} = 98.9802708766234$$
$$x_{14} = 11.1656732464925$$
$$x_{15} = 5.06026836518236$$
$$x_{16} = -89.5578448748316$$
$$x_{17} = -8.1119826164815$$
$$x_{18} = -33.0481076541221$$
$$x_{19} = -64.4339234092994$$
$$x_{20} = -14.2814705578331$$
$$x_{21} = -76.9951606189843$$
$$x_{22} = 23.6446256872296$$
$$x_{23} = -39.3213891805148$$
$$x_{24} = -70.7143126702811$$
$$x_{25} = -51.8751921133361$$
$$x_{26} = 67.5736151455506$$
$$x_{27} = -158.663073635328$$
$$x_{28} = 36.1827931655659$$
$$x_{29} = 61.2934134764651$$
$$x_{30} = 80.1354115339569$$
$$x_{31} = -20.519983804286$$
$$x_{32} = 48.7352892631046$$
$$x_{33} = 73.8543203258377$$
$$x_{34} = 17.3901768408315$$
Puntos máximos de la función:
$$x_{34} = -48.7361295392441$$
$$x_{34} = 20.5152887094394$$
$$x_{34} = -42.4591369509051$$
$$x_{34} = -92.6986728032932$$
$$x_{34} = -23.6481708154477$$
$$x_{34} = 58.1535518920027$$
$$x_{34} = -67.5740526651947$$
$$x_{34} = 14.2718933831386$$
$$x_{34} = 2.22109820882798$$
$$x_{34} = 64.4334422583637$$
$$x_{34} = -11.1810751482291$$
$$x_{34} = -0.649755168374399$$
$$x_{34} = 45.5964586594414$$
$$x_{34} = -55.0145455295839$$
$$x_{34} = 26.7767536014234$$
$$x_{34} = 51.8744502729041$$
$$x_{34} = -17.3966801527861$$
$$x_{34} = 33.046284696792$$
$$x_{34} = 39.3200997911221$$
$$x_{34} = 89.5575956718273$$
$$x_{34} = -29.9130423127956$$
$$x_{34} = 76.9948235344069$$
$$x_{34} = 95.8393334645179$$
$$x_{34} = -36.1843149453611$$
$$x_{34} = 70.7139131086823$$
$$x_{34} = -80.1357227352106$$
$$x_{34} = 83.276074800594$$
$$x_{34} = -86.4170728188749$$
$$x_{34} = -5.12485035716986$$
$$x_{34} = 8.08365076227732$$
$$x_{34} = -98.9804749142348$$
$$x_{34} = -61.2939451205206$$
$$x_{34} = -73.854686660561$$
Decrece en los intervalos
$$\left[98.9802708766234, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -158.663073635328\right]$$
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
$$- x \left(x + 1\right) \sin{\left(x \right)} + 2 \left(2 x + 1\right) \cos{\left(x \right)} + 2 \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- x \left(x + 1\right) \sin{\left(x \right)} + 2 \left(2 x + 1\right) \cos{\left(x \right)} + 2 \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(x^{2} + x\right) \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\left(x^{2} + x\right) \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\left(x^{2} + x\right) \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\left(x^{2} + x\right) \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x^2 + x)*sin(x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x^{2} + x\right) \sin{\left(x \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle x$$
$$\lim_{x \to \infty}\left(\frac{\left(x^{2} + x\right) \sin{\left(x \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle x$$
$$\lim_{x \to -\infty}\left(\frac{\left(x^{2} + x\right) \sin{\left(x \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle x$$
$$\lim_{x \to \infty}\left(\frac{\left(x^{2} + x\right) \sin{\left(x \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle x$$
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:
$$\left(x^{2} + x\right) \sin{\left(x \right)} = - \left(x^{2} - x\right) \sin{\left(x \right)}$$
- No
$$\left(x^{2} + x\right) \sin{\left(x \right)} = \left(x^{2} - x\right) \sin{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(x^{2} + x\right) \sin{\left(x \right)} = - \left(x^{2} - x\right) \sin{\left(x \right)}$$
- No
$$\left(x^{2} + x\right) \sin{\left(x \right)} = \left(x^{2} - x\right) \sin{\left(x \right)}$$
- No
es decir, función
no es
par ni impar