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Resolución de integrales/ (sin(x)*(sin(x)+2cos(x)))/5xsin^2(x)+4

Integral de (sin(x)*(sin(x)+2cos(x)))/5xsin^2(x)+4 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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 |  /sin(x)*(sin(x) + 2*cos(x))      2       \   
 |  |--------------------------*x*sin (x) + 4| dx
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0
∫01(x(sin⁡(x)+2cos⁡(x))sin⁡(x)5sin⁡2(x)+4) dx\int\limits_{0}^{1} \left(x \frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) \sin{\left(x \right)}}{5} \sin^{2}{\left(x \right)} + 4\right)\, dx 
Integral((((sin(x)*(sin(x) + 2*cos(x)))/5)*x)*sin(x)^2 + 4, (x, 0, 1))
Solución detallada

  1. Integramos término a término:

    1. Vuelva a escribir el integrando:

      x(sin⁡(x)+2cos⁡(x))sin⁡(x)5sin⁡2(x)=xsin⁡4(x)5+2xsin⁡3(x)cos⁡(x)5x \frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) \sin{\left(x \right)}}{5} \sin^{2}{\left(x \right)} = \frac{x \sin^{4}{\left(x \right)}}{5} + \frac{2 x \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{5}

    2. Integramos término a término:

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫xsin⁡4(x)5 dx=∫xsin⁡4(x) dx5\int \frac{x \sin^{4}{\left(x \right)}}{5}\, dx = \frac{\int x \sin^{4}{\left(x \right)}\, dx}{5}

        1. No puedo encontrar los pasos en la búsqueda de esta integral.

          Pero la integral

          3x2sin⁡4(x)16+3x2sin⁡2(x)cos⁡2(x)8+3x2cos⁡4(x)16−5xsin⁡3(x)cos⁡(x)8−3xsin⁡(x)cos⁡3(x)8+5sin⁡4(x)32−3cos⁡4(x)32\frac{3 x^{2} \sin^{4}{\left(x \right)}}{16} + \frac{3 x^{2} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{8} + \frac{3 x^{2} \cos^{4}{\left(x \right)}}{16} - \frac{5 x \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{8} - \frac{3 x \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{8} + \frac{5 \sin^{4}{\left(x \right)}}{32} - \frac{3 \cos^{4}{\left(x \right)}}{32}

        Por lo tanto, el resultado es: 3x2sin⁡4(x)80+3x2sin⁡2(x)cos⁡2(x)40+3x2cos⁡4(x)80−xsin⁡3(x)cos⁡(x)8−3xsin⁡(x)cos⁡3(x)40+sin⁡4(x)32−3cos⁡4(x)160\frac{3 x^{2} \sin^{4}{\left(x \right)}}{80} + \frac{3 x^{2} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{40} + \frac{3 x^{2} \cos^{4}{\left(x \right)}}{80} - \frac{x \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{8} - \frac{3 x \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{40} + \frac{\sin^{4}{\left(x \right)}}{32} - \frac{3 \cos^{4}{\left(x \right)}}{160}

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫2xsin⁡3(x)cos⁡(x)5 dx=2∫xsin⁡3(x)cos⁡(x) dx5\int \frac{2 x \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{5}\, dx = \frac{2 \int x \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx}{5}

        1. No puedo encontrar los pasos en la búsqueda de esta integral.

          Pero la integral

          5xsin⁡4(x)32−3xsin⁡2(x)cos⁡2(x)16−3xcos⁡4(x)32+5sin⁡3(x)cos⁡(x)32+3sin⁡(x)cos⁡3(x)32\frac{5 x \sin^{4}{\left(x \right)}}{32} - \frac{3 x \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{16} - \frac{3 x \cos^{4}{\left(x \right)}}{32} + \frac{5 \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{32} + \frac{3 \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{32}

        Por lo tanto, el resultado es: xsin⁡4(x)16−3xsin⁡2(x)cos⁡2(x)40−3xcos⁡4(x)80+sin⁡3(x)cos⁡(x)16+3sin⁡(x)cos⁡3(x)80\frac{x \sin^{4}{\left(x \right)}}{16} - \frac{3 x \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{40} - \frac{3 x \cos^{4}{\left(x \right)}}{80} + \frac{\sin^{3}{\left(x \right)} \cos{\left(x \right)}}{16} + \frac{3 \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{80}

      El resultado es: 3x2sin⁡4(x)80+3x2sin⁡2(x)cos⁡2(x)40+3x2cos⁡4(x)80+xsin⁡4(x)16−xsin⁡3(x)cos⁡(x)8−3xsin⁡2(x)cos⁡2(x)40−3xsin⁡(x)cos⁡3(x)40−3xcos⁡4(x)80+sin⁡4(x)32+sin⁡3(x)cos⁡(x)16+3sin⁡(x)cos⁡3(x)80−3cos⁡4(x)160\frac{3 x^{2} \sin^{4}{\left(x \right)}}{80} + \frac{3 x^{2} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{40} + \frac{3 x^{2} \cos^{4}{\left(x \right)}}{80} + \frac{x \sin^{4}{\left(x \right)}}{16} - \frac{x \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{8} - \frac{3 x \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{40} - \frac{3 x \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{40} - \frac{3 x \cos^{4}{\left(x \right)}}{80} + \frac{\sin^{4}{\left(x \right)}}{32} + \frac{\sin^{3}{\left(x \right)} \cos{\left(x \right)}}{16} + \frac{3 \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{80} - \frac{3 \cos^{4}{\left(x \right)}}{160}

    1. La integral de las constantes tienen esta constante multiplicada por la variable de integración:

      ∫4 dx=4x\int 4\, dx = 4 x

    El resultado es: 3x2sin⁡4(x)80+3x2sin⁡2(x)cos⁡2(x)40+3x2cos⁡4(x)80+xsin⁡4(x)16−xsin⁡3(x)cos⁡(x)8−3xsin⁡2(x)cos⁡2(x)40−3xsin⁡(x)cos⁡3(x)40−3xcos⁡4(x)80+4x+sin⁡4(x)32+sin⁡3(x)cos⁡(x)16+3sin⁡(x)cos⁡3(x)80−3cos⁡4(x)160\frac{3 x^{2} \sin^{4}{\left(x \right)}}{80} + \frac{3 x^{2} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{40} + \frac{3 x^{2} \cos^{4}{\left(x \right)}}{80} + \frac{x \sin^{4}{\left(x \right)}}{16} - \frac{x \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{8} - \frac{3 x \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{40} - \frac{3 x \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{40} - \frac{3 x \cos^{4}{\left(x \right)}}{80} + 4 x + \frac{\sin^{4}{\left(x \right)}}{32} + \frac{\sin^{3}{\left(x \right)} \cos{\left(x \right)}}{16} + \frac{3 \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{80} - \frac{3 \cos^{4}{\left(x \right)}}{160}

  2. Ahora simplificar:

    3x280+xsin⁡4(x)10−xsin⁡(2x)20+xsin⁡(4x)160+317x80+sin⁡4(x)80+sin⁡(2x)40−sin⁡(4x)320−3cos⁡(2x)160\frac{3 x^{2}}{80} + \frac{x \sin^{4}{\left(x \right)}}{10} - \frac{x \sin{\left(2 x \right)}}{20} + \frac{x \sin{\left(4 x \right)}}{160} + \frac{317 x}{80} + \frac{\sin^{4}{\left(x \right)}}{80} + \frac{\sin{\left(2 x \right)}}{40} - \frac{\sin{\left(4 x \right)}}{320} - \frac{3 \cos{\left(2 x \right)}}{160}

  3. Añadimos la constante de integración:

    3x280+xsin⁡4(x)10−xsin⁡(2x)20+xsin⁡(4x)160+317x80+sin⁡4(x)80+sin⁡(2x)40−sin⁡(4x)320−3cos⁡(2x)160+constant\frac{3 x^{2}}{80} + \frac{x \sin^{4}{\left(x \right)}}{10} - \frac{x \sin{\left(2 x \right)}}{20} + \frac{x \sin{\left(4 x \right)}}{160} + \frac{317 x}{80} + \frac{\sin^{4}{\left(x \right)}}{80} + \frac{\sin{\left(2 x \right)}}{40} - \frac{\sin{\left(4 x \right)}}{320} - \frac{3 \cos{\left(2 x \right)}}{160}+ \mathrm{constant}

Respuesta:

3x280+xsin⁡4(x)10−xsin⁡(2x)20+xsin⁡(4x)160+317x80+sin⁡4(x)80+sin⁡(2x)40−sin⁡(4x)320−3cos⁡(2x)160+constant\frac{3 x^{2}}{80} + \frac{x \sin^{4}{\left(x \right)}}{10} - \frac{x \sin{\left(2 x \right)}}{20} + \frac{x \sin{\left(4 x \right)}}{160} + \frac{317 x}{80} + \frac{\sin^{4}{\left(x \right)}}{80} + \frac{\sin{\left(2 x \right)}}{40} - \frac{\sin{\left(4 x \right)}}{320} - \frac{3 \cos{\left(2 x \right)}}{160}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                                                                                                                                                                                              
 |                                                                4         4             4           4         3                2    4         2    4           3                    2       2             3                  3                2    2       2   
 | /sin(x)*(sin(x) + 2*cos(x))      2       \                3*cos (x)   sin (x)   3*x*cos (x)   x*sin (x)   sin (x)*cos(x)   3*x *cos (x)   3*x *sin (x)   3*cos (x)*sin(x)   3*x*cos (x)*sin (x)   3*x*cos (x)*sin(x)   x*sin (x)*cos(x)   3*x *cos (x)*sin (x)
 | |--------------------------*x*sin (x) + 4| dx = C + 4*x - --------- + ------- - ----------- + --------- + -------------- + ------------ + ------------ + ---------------- - ------------------- - ------------------ - ---------------- + --------------------
 | \            5                           /                   160         32          80           16            16              80             80               80                   40                   40                  8                    40         
 |                                                                                                                                                                                                                                                               
/
∫(x(sin⁡(x)+2cos⁡(x))sin⁡(x)5sin⁡2(x)+4) dx=C+3x2sin⁡4(x)80+3x2sin⁡2(x)cos⁡2(x)40+3x2cos⁡4(x)80+xsin⁡4(x)16−xsin⁡3(x)cos⁡(x)8−3xsin⁡2(x)cos⁡2(x)40−3xsin⁡(x)cos⁡3(x)40−3xcos⁡4(x)80+4x+sin⁡4(x)32+sin⁡3(x)cos⁡(x)16+3sin⁡(x)cos⁡3(x)80−3cos⁡4(x)160\int \left(x \frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) \sin{\left(x \right)}}{5} \sin^{2}{\left(x \right)} + 4\right)\, dx = C + \frac{3 x^{2} \sin^{4}{\left(x \right)}}{80} + \frac{3 x^{2} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{40} + \frac{3 x^{2} \cos^{4}{\left(x \right)}}{80} + \frac{x \sin^{4}{\left(x \right)}}{16} - \frac{x \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{8} - \frac{3 x \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{40} - \frac{3 x \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{40} - \frac{3 x \cos^{4}{\left(x \right)}}{80} + 4 x + \frac{\sin^{4}{\left(x \right)}}{32} + \frac{\sin^{3}{\left(x \right)} \cos{\left(x \right)}}{16} + \frac{3 \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{80} - \frac{3 \cos^{4}{\left(x \right)}}{160}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.905-5
Trazar el gráfico f(x)
Respuesta [src] 
           4            4           3                3          
643   3*cos (1)   21*sin (1)   3*cos (1)*sin(1)   sin (1)*cos(1)
--- - --------- + ---------- - ---------------- - --------------
160      160         160              80                16
−sin⁡3(1)cos⁡(1)16−3sin⁡(1)cos⁡3(1)80−3cos⁡4(1)160+21sin⁡4(1)160+643160- \frac{\sin^{3}{\left(1 \right)} \cos{\left(1 \right)}}{16} - \frac{3 \sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{80} - \frac{3 \cos^{4}{\left(1 \right)}}{160} + \frac{21 \sin^{4}{\left(1 \right)}}{160} + \frac{643}{160} 
=
= 
           4            4           3                3          
643   3*cos (1)   21*sin (1)   3*cos (1)*sin(1)   sin (1)*cos(1)
--- - --------- + ---------- - ---------------- - --------------
160      160         160              80                16
−sin⁡3(1)cos⁡(1)16−3sin⁡(1)cos⁡3(1)80−3cos⁡4(1)160+21sin⁡4(1)160+643160- \frac{\sin^{3}{\left(1 \right)} \cos{\left(1 \right)}}{16} - \frac{3 \sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{80} - \frac{3 \cos^{4}{\left(1 \right)}}{160} + \frac{21 \sin^{4}{\left(1 \right)}}{160} + \frac{643}{160} 
643/160 - 3*cos(1)^4/160 + 21*sin(1)^4/160 - 3*cos(1)^3*sin(1)/80 - sin(1)^3*cos(1)/16
Respuesta numérica [src] 
4.05785920585416
4.05785920585416

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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