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Resolución de integrales/ x^3+2/ cos(n*x)/ 3*x^3/ (0,3*x^3+2)*cos(n*x)

Integral de (0,3*x^3+2)*cos(n*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                       
  /                       
 |                        
 |  /   3    \            
 |  |3*x     |            
 |  |---- + 2|*cos(n*x) dx
 |  \ 10     /            
 |                        
/                         
0
∫01(3x310+2)cos⁡(nx) dx\int\limits_{0}^{1} \left(\frac{3 x^{3}}{10} + 2\right) \cos{\left(n x \right)}\, dx 
Integral((3*x^3/10 + 2)*cos(n*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                             //                          4                                    \                              
                                                             ||                         x                                     |                              
                                                             ||                         --                           for n = 0|                              
                                                             ||                         4                                     |                              
                                                             ||                                                               |                              
                                                             ||/              2                                               |                              
                                                             |||2*cos(n*x)   x *cos(n*x)   2*x*sin(n*x)                       |                              
                                                           9*|<|---------- - ----------- + ------------  for n != 0           |                              
                                                             ||<     3            n              2                            |                              
                                                             |||    n                           n                             |                              
                                                             |||                                                              |        //   x      for n = 0\
  /                                                          ||\                   0                     otherwise            |      3 ||                   |
 |                                                           ||----------------------------------------------------  otherwise|   3*x *|
∫(3x310+2)cos⁡(nx) dx=C+3x3({xfor n=0sin⁡(nx)notherwise)10+2({xfor n=0sin⁡(nx)notherwise)−9({x44for n=0{−x2cos⁡(nx)n+2xsin⁡(nx)n2+2cos⁡(nx)n3for n≠00otherwisenotherwise)10\int \left(\frac{3 x^{3}}{10} + 2\right) \cos{\left(n x \right)}\, dx = C + \frac{3 x^{3} \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right)}{10} + 2 \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - \frac{9 \left(\begin{cases} \frac{x^{4}}{4} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{x^{2} \cos{\left(n x \right)}}{n} + \frac{2 x \sin{\left(n x \right)}}{n^{2}} + \frac{2 \cos{\left(n x \right)}}{n^{3}} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right)}{10}
Simplificar
Respuesta [src] 
/ 9     9*cos(n)   9*sin(n)   9*cos(n)   23*sin(n)                                  
|---- - -------- - -------- + -------- + ---------  for And(n > -oo, n < oo, n != 0)
|   4        4          3          2        10*n                                    
|5*n      5*n        5*n       10*n                                                 
<                                                                                   
|                       83                                                          
|                       --                                     otherwise            
|                       40                                                          
\
{23sin⁡(n)10n+9cos⁡(n)10n2−9sin⁡(n)5n3−9cos⁡(n)5n4+95n4for n>−∞∧n<∞∧n≠08340otherwise\begin{cases} \frac{23 \sin{\left(n \right)}}{10 n} + \frac{9 \cos{\left(n \right)}}{10 n^{2}} - \frac{9 \sin{\left(n \right)}}{5 n^{3}} - \frac{9 \cos{\left(n \right)}}{5 n^{4}} + \frac{9}{5 n^{4}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{83}{40} & \text{otherwise} \end{cases} 
=
= 
/ 9     9*cos(n)   9*sin(n)   9*cos(n)   23*sin(n)                                  
|---- - -------- - -------- + -------- + ---------  for And(n > -oo, n < oo, n != 0)
|   4        4          3          2        10*n                                    
|5*n      5*n        5*n       10*n                                                 
<                                                                                   
|                       83                                                          
|                       --                                     otherwise            
|                       40                                                          
\
{23sin⁡(n)10n+9cos⁡(n)10n2−9sin⁡(n)5n3−9cos⁡(n)5n4+95n4for n>−∞∧n<∞∧n≠08340otherwise\begin{cases} \frac{23 \sin{\left(n \right)}}{10 n} + \frac{9 \cos{\left(n \right)}}{10 n^{2}} - \frac{9 \sin{\left(n \right)}}{5 n^{3}} - \frac{9 \cos{\left(n \right)}}{5 n^{4}} + \frac{9}{5 n^{4}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{83}{40} & \text{otherwise} \end{cases} 
Piecewise((9/(5*n^4) - 9*cos(n)/(5*n^4) - 9*sin(n)/(5*n^3) + 9*cos(n)/(10*n^2) + 23*sin(n)/(10*n), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (83/40, True))

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

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