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Resolución de integrales/ (x-6)/ cos(x-6)/4+sin2(x-6)

Integral de cos(x-6)/4+sin2(x-6) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                              
  /                              
 |                               
 |  /cos(x - 6)      2       \   
 |  |---------- + sin (x - 6)| dx
 |  \    4                   /   
 |                               
/                                
0
∫01(sin⁡2(x−6)+cos⁡(x−6)4) dx\int\limits_{0}^{1} \left(\sin^{2}{\left(x - 6 \right)} + \frac{\cos{\left(x - 6 \right)}}{4}\right)\, dx 
Integral(cos(x - 6)/4 + sin(x - 6)^2, (x, 0, 1))
Solución detallada

  1. Integramos término a término:

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

      Pero la integral

      xtan⁡4(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2+2xtan⁡2(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2+x2tan⁡4(x2−3)+4tan⁡2(x2−3)+2+2tan⁡3(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2−2tan⁡(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2\frac{x \tan^{4}{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} + \frac{2 x \tan^{2}{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} + \frac{x}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} + \frac{2 \tan^{3}{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} - \frac{2 \tan{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2}

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

      ∫cos⁡(x−6)4 dx=∫cos⁡(x−6) dx4\int \frac{\cos{\left(x - 6 \right)}}{4}\, dx = \frac{\int \cos{\left(x - 6 \right)}\, dx}{4}

      1. que u=x−6u = x - 6.

        Luego que du=dxdu = dx y ponemos dudu:

        ∫cos⁡(u) du\int \cos{\left(u \right)}\, du

        1. La integral del coseno es seno:

          ∫cos⁡(u) du=sin⁡(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

        Si ahora sustituir uu más en:

        sin⁡(x−6)\sin{\left(x - 6 \right)}

      Por lo tanto, el resultado es: sin⁡(x−6)4\frac{\sin{\left(x - 6 \right)}}{4}

    El resultado es: xtan⁡4(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2+2xtan⁡2(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2+x2tan⁡4(x2−3)+4tan⁡2(x2−3)+2+sin⁡(x−6)4+2tan⁡3(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2−2tan⁡(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2\frac{x \tan^{4}{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} + \frac{2 x \tan^{2}{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} + \frac{x}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} + \frac{\sin{\left(x - 6 \right)}}{4} + \frac{2 \tan^{3}{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} - \frac{2 \tan{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2}

  2. Ahora simplificar:

    xtan⁡4(x2−3)2+xtan⁡2(x2−3)+x2+(tan⁡4(x2−3)+2tan⁡2(x2−3)+1)sin⁡(x−6)4+tan⁡3(x2−3)−tan⁡(x2−3)tan⁡4(x2−3)+2tan⁡2(x2−3)+1\frac{\frac{x \tan^{4}{\left(\frac{x}{2} - 3 \right)}}{2} + x \tan^{2}{\left(\frac{x}{2} - 3 \right)} + \frac{x}{2} + \frac{\left(\tan^{4}{\left(\frac{x}{2} - 3 \right)} + 2 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 1\right) \sin{\left(x - 6 \right)}}{4} + \tan^{3}{\left(\frac{x}{2} - 3 \right)} - \tan{\left(\frac{x}{2} - 3 \right)}}{\tan^{4}{\left(\frac{x}{2} - 3 \right)} + 2 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 1}

  3. Añadimos la constante de integración:

    xtan⁡4(x2−3)2+xtan⁡2(x2−3)+x2+(tan⁡4(x2−3)+2tan⁡2(x2−3)+1)sin⁡(x−6)4+tan⁡3(x2−3)−tan⁡(x2−3)tan⁡4(x2−3)+2tan⁡2(x2−3)+1+constant\frac{\frac{x \tan^{4}{\left(\frac{x}{2} - 3 \right)}}{2} + x \tan^{2}{\left(\frac{x}{2} - 3 \right)} + \frac{x}{2} + \frac{\left(\tan^{4}{\left(\frac{x}{2} - 3 \right)} + 2 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 1\right) \sin{\left(x - 6 \right)}}{4} + \tan^{3}{\left(\frac{x}{2} - 3 \right)} - \tan{\left(\frac{x}{2} - 3 \right)}}{\tan^{4}{\left(\frac{x}{2} - 3 \right)} + 2 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 1}+ \mathrm{constant}

Respuesta:

xtan⁡4(x2−3)2+xtan⁡2(x2−3)+x2+(tan⁡4(x2−3)+2tan⁡2(x2−3)+1)sin⁡(x−6)4+tan⁡3(x2−3)−tan⁡(x2−3)tan⁡4(x2−3)+2tan⁡2(x2−3)+1+constant\frac{\frac{x \tan^{4}{\left(\frac{x}{2} - 3 \right)}}{2} + x \tan^{2}{\left(\frac{x}{2} - 3 \right)} + \frac{x}{2} + \frac{\left(\tan^{4}{\left(\frac{x}{2} - 3 \right)} + 2 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 1\right) \sin{\left(x - 6 \right)}}{4} + \tan^{3}{\left(\frac{x}{2} - 3 \right)} - \tan{\left(\frac{x}{2} - 3 \right)}}{\tan^{4}{\left(\frac{x}{2} - 3 \right)} + 2 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 1}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                                       /     x\                              3/     x\                             4/     x\                              2/     x\         
 |                                                                                                   2*tan|-3 + -|                         2*tan |-3 + -|                        x*tan |-3 + -|                       2*x*tan |-3 + -|         
 | /cos(x - 6)      2       \          sin(x - 6)                    x                                    \     2/                               \     2/                              \     2/                               \     2/         
 | |---------- + sin (x - 6)| dx = C + ---------- + ----------------------------------- - ----------------------------------- + ----------------------------------- + ----------------------------------- + -----------------------------------
 | \    4                   /              4                 4/     x\        2/     x\            4/     x\        2/     x\            4/     x\        2/     x\            4/     x\        2/     x\            4/     x\        2/     x\
 |                                                  2 + 2*tan |-3 + -| + 4*tan |-3 + -|   2 + 2*tan |-3 + -| + 4*tan |-3 + -|   2 + 2*tan |-3 + -| + 4*tan |-3 + -|   2 + 2*tan |-3 + -| + 4*tan |-3 + -|   2 + 2*tan |-3 + -| + 4*tan |-3 + -|
/                                                             \     2/         \     2/             \     2/         \     2/             \     2/         \     2/             \     2/         \     2/             \     2/         \     2/
∫(sin⁡2(x−6)+cos⁡(x−6)4) dx=C+xtan⁡4(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2+2xtan⁡2(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2+x2tan⁡4(x2−3)+4tan⁡2(x2−3)+2+sin⁡(x−6)4+2tan⁡3(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2−2tan⁡(x2−3)2tan⁡4(x2−3)+4tan⁡2(x2−3)+2\int \left(\sin^{2}{\left(x - 6 \right)} + \frac{\cos{\left(x - 6 \right)}}{4}\right)\, dx = C + \frac{x \tan^{4}{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} + \frac{2 x \tan^{2}{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} + \frac{x}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} + \frac{\sin{\left(x - 6 \right)}}{4} + \frac{2 \tan^{3}{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2} - \frac{2 \tan{\left(\frac{x}{2} - 3 \right)}}{2 \tan^{4}{\left(\frac{x}{2} - 3 \right)} + 4 \tan^{2}{\left(\frac{x}{2} - 3 \right)} + 2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.902-1
Trazar el gráfico f(x)
Respuesta [src] 
   2         2                                                     
cos (5)   sin (5)   sin(5)   sin(6)   cos(5)*sin(5)   cos(6)*sin(6)
------- + ------- - ------ + ------ + ------------- - -------------
   2         2        4        4            2               2
sin⁡(5)cos⁡(5)2+sin⁡(6)4+cos⁡2(5)2−sin⁡(6)cos⁡(6)2−sin⁡(5)4+sin⁡2(5)2\frac{\sin{\left(5 \right)} \cos{\left(5 \right)}}{2} + \frac{\sin{\left(6 \right)}}{4} + \frac{\cos^{2}{\left(5 \right)}}{2} - \frac{\sin{\left(6 \right)} \cos{\left(6 \right)}}{2} - \frac{\sin{\left(5 \right)}}{4} + \frac{\sin^{2}{\left(5 \right)}}{2} 
=
= 
   2         2                                                     
cos (5)   sin (5)   sin(5)   sin(6)   cos(5)*sin(5)   cos(6)*sin(6)
------- + ------- - ------ + ------ + ------------- - -------------
   2         2        4        4            2               2
sin⁡(5)cos⁡(5)2+sin⁡(6)4+cos⁡2(5)2−sin⁡(6)cos⁡(6)2−sin⁡(5)4+sin⁡2(5)2\frac{\sin{\left(5 \right)} \cos{\left(5 \right)}}{2} + \frac{\sin{\left(6 \right)}}{4} + \frac{\cos^{2}{\left(5 \right)}}{2} - \frac{\sin{\left(6 \right)} \cos{\left(6 \right)}}{2} - \frac{\sin{\left(5 \right)}}{4} + \frac{\sin^{2}{\left(5 \right)}}{2} 
cos(5)^2/2 + sin(5)^2/2 - sin(5)/4 + sin(6)/4 + cos(5)*sin(5)/2 - cos(6)*sin(6)/2
Respuesta numérica [src] 
0.668015145893819
0.668015145893819

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

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