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Resolución de integrales/ i*n*x/ sin3x/ sin3x*cos((pi*n*x)/4)

Integral de sin3x*cos((pi*n*x)/4) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                        
  /                        
 |                         
 |              /pi*n*x\   
 |  sin(3*x)*cos|------| dx
 |              \  4   /   
 |                         
/                          
0
∫01sin⁡(3x)cos⁡(xπn4) dx\int\limits_{0}^{1} \sin{\left(3 x \right)} \cos{\left(\frac{x \pi n}{4} \right)}\, dx 
Integral(sin(3*x)*cos(((pi*n)*x)/4), (x, 0, 1))
Respuesta [src] 
/                                 2                                                       
|                          1   cos (3)                                  /    -12       12\
|                          - - -------                            for Or|n = ----, n = --|
|                          6      6                                     \     pi       pi/
|                                                                                         
<                               /pi*n\                    /pi*n\                          
|                  48*cos(3)*cos|----|   4*pi*n*sin(3)*sin|----|                          
|        48                     \ 4  /                    \ 4  /                          
|- ------------- + ------------------- + -----------------------         otherwise        
|           2  2               2  2                    2  2                               
\  -144 + pi *n       -144 + pi *n            -144 + pi *n
{16−cos⁡2(3)6for n=−12π∨n=12π4πnsin⁡(3)sin⁡(πn4)π2n2−144+48cos⁡(3)cos⁡(πn4)π2n2−144−48π2n2−144otherwise\begin{cases} \frac{1}{6} - \frac{\cos^{2}{\left(3 \right)}}{6} & \text{for}\: n = - \frac{12}{\pi} \vee n = \frac{12}{\pi} \\\frac{4 \pi n \sin{\left(3 \right)} \sin{\left(\frac{\pi n}{4} \right)}}{\pi^{2} n^{2} - 144} + \frac{48 \cos{\left(3 \right)} \cos{\left(\frac{\pi n}{4} \right)}}{\pi^{2} n^{2} - 144} - \frac{48}{\pi^{2} n^{2} - 144} & \text{otherwise} \end{cases} 
=
= 
/                                 2                                                       
|                          1   cos (3)                                  /    -12       12\
|                          - - -------                            for Or|n = ----, n = --|
|                          6      6                                     \     pi       pi/
|                                                                                         
<                               /pi*n\                    /pi*n\                          
|                  48*cos(3)*cos|----|   4*pi*n*sin(3)*sin|----|                          
|        48                     \ 4  /                    \ 4  /                          
|- ------------- + ------------------- + -----------------------         otherwise        
|           2  2               2  2                    2  2                               
\  -144 + pi *n       -144 + pi *n            -144 + pi *n
{16−cos⁡2(3)6for n=−12π∨n=12π4πnsin⁡(3)sin⁡(πn4)π2n2−144+48cos⁡(3)cos⁡(πn4)π2n2−144−48π2n2−144otherwise\begin{cases} \frac{1}{6} - \frac{\cos^{2}{\left(3 \right)}}{6} & \text{for}\: n = - \frac{12}{\pi} \vee n = \frac{12}{\pi} \\\frac{4 \pi n \sin{\left(3 \right)} \sin{\left(\frac{\pi n}{4} \right)}}{\pi^{2} n^{2} - 144} + \frac{48 \cos{\left(3 \right)} \cos{\left(\frac{\pi n}{4} \right)}}{\pi^{2} n^{2} - 144} - \frac{48}{\pi^{2} n^{2} - 144} & \text{otherwise} \end{cases} 
Piecewise((1/6 - cos(3)^2/6, (n = -12/pi)∨(n = 12/pi)), (-48/(-144 + pi^2*n^2) + 48*cos(3)*cos(pi*n/4)/(-144 + pi^2*n^2) + 4*pi*n*sin(3)*sin(pi*n/4)/(-144 + pi^2*n^2), True))

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

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