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Resolución de integrales/ (2x+3)/ (2x+3)*cos((2pi*x)/3n)

Integral de (2x+3)*cos((2pi*x)/3n) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                           
  /                           
 |                            
 |               /2*pi*x  \   
 |  (2*x + 3)*cos|------*n| dx
 |               \  3     /   
 |                            
/                             
0
∫01(2x+3)cos⁡(n2πx3) dx\int\limits_{0}^{1} \left(2 x + 3\right) \cos{\left(n \frac{2 \pi x}{3} \right)}\, dx 
Integral((2*x + 3)*cos((((2*pi)*x)/3)*n), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                      //                   2                             \                                                                      
                                      ||                  x                              |                                                                      
                                      ||                  --                    for n = 0|                                                                      
                                      ||                  2                              |                                                                      
  /                                   ||                                                 |     //       x         for n = 0\       //       x         for n = 0\
 |                                    ||  //      /2*pi*n*x\                 \           |     ||                          |       ||                          |
 |              /2*pi*x  \            ||  ||-3*cos|--------|                 |           |     ||     /2*pi*n*x\           |       ||     /2*pi*n*x\           |
 | (2*x + 3)*cos|------*n| dx = C - 2*|<  ||      \   3    /      2*pi*n     |           | + 3*|<3*sin|--------|           | + 2*x*|<3*sin|--------|           |
 |              \  3     /            ||3*|<----------------  for ------ != 0|           |     ||     \   3    /           |       ||     \   3    /           |
 |                                    ||  ||     2*pi*n             3        |           |     ||---------------  otherwise|       ||---------------  otherwise|
/                                     ||  ||                                 |           |     \\     2*pi*n               /       \\     2*pi*n               /
                                      ||  \\       0             otherwise   /           |                                                                      
                                      ||--------------------------------------  otherwise|                                                                      
                                      ||                2*pi*n                           |                                                                      
                                      \\                                                 /
∫(2x+3)cos⁡(n2πx3) dx=C+2x({xfor n=03sin⁡(2πnx3)2πnotherwise)+3({xfor n=03sin⁡(2πnx3)2πnotherwise)−2({x22for n=03({−3cos⁡(2πnx3)2πnfor 2πn3≠00otherwise)2πnotherwise)\int \left(2 x + 3\right) \cos{\left(n \frac{2 \pi x}{3} \right)}\, dx = C + 2 x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{3 \sin{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right) + 3 \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{3 \sin{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{3 \left(\begin{cases} - \frac{3 \cos{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{for}\: \frac{2 \pi n}{3} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{2 \pi n} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/                  /2*pi*n\         /2*pi*n\                                  
|             9*cos|------|   15*sin|------|                                  
|     9            \  3   /         \  3   /                                  
|- -------- + ------------- + --------------  for And(n > -oo, n < oo, n != 0)
<      2  2          2  2         2*pi*n                                      
|  2*pi *n       2*pi *n                                                      
|                                                                             
|                     4                                  otherwise            
\
{15sin⁡(2πn3)2πn+9cos⁡(2πn3)2π2n2−92π2n2for n>−∞∧n<∞∧n≠04otherwise\begin{cases} \frac{15 \sin{\left(\frac{2 \pi n}{3} \right)}}{2 \pi n} + \frac{9 \cos{\left(\frac{2 \pi n}{3} \right)}}{2 \pi^{2} n^{2}} - \frac{9}{2 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\4 & \text{otherwise} \end{cases} 
=
= 
/                  /2*pi*n\         /2*pi*n\                                  
|             9*cos|------|   15*sin|------|                                  
|     9            \  3   /         \  3   /                                  
|- -------- + ------------- + --------------  for And(n > -oo, n < oo, n != 0)
<      2  2          2  2         2*pi*n                                      
|  2*pi *n       2*pi *n                                                      
|                                                                             
|                     4                                  otherwise            
\
{15sin⁡(2πn3)2πn+9cos⁡(2πn3)2π2n2−92π2n2for n>−∞∧n<∞∧n≠04otherwise\begin{cases} \frac{15 \sin{\left(\frac{2 \pi n}{3} \right)}}{2 \pi n} + \frac{9 \cos{\left(\frac{2 \pi n}{3} \right)}}{2 \pi^{2} n^{2}} - \frac{9}{2 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\4 & \text{otherwise} \end{cases} 
Piecewise((-9/(2*pi^2*n^2) + 9*cos(2*pi*n/3)/(2*pi^2*n^2) + 15*sin(2*pi*n/3)/(2*pi*n), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (4, True))

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

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