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Resolución de integrales/ sin(pinx/2)

Integral de sin(pinx/2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                 
  /                 
 |                  
 |     /p*log(x)\   
 |  sin|--------| dx
 |     \   2    /   
 |                  
/                   
0
∫0πsin⁡(plog⁡(x)2) dx\int\limits_{0}^{\pi} \sin{\left(\frac{p \log{\left(x \right)}}{2} \right)}\, dx 
Integral(sin((p*log(x))/2), (x, 0, pi))
Respuesta (Indefinida) [src] 
                          //                                                                              2                                                                              \
                          ||                                                                I*log(x)   I*x                                                                               |
                          ||                                                                -------- - ----                                                                  for p = -2*I|
                          ||                                                                   2        4                                                                                |
                          ||                                                                                                                                                             |
  /                       ||                                                                               2                                                                             |
 |                        ||                                                                 I*log(x)   I*x                                                                              |
 |    /p*log(x)\          ||                                                               - -------- + ----                                                                 for p = 2*I |
 | sin|--------| dx = C + |<                                                                    2        4                                                                               |
 |    \   2    /          ||                                                                                                                                                             |
 |                        ||                                                                       /p*log(x)\                                       2/p*log(x)\                          |
/                         ||                                                                8*x*tan|--------|                              2*p*x*tan |--------|                          |
                          ||                      2*p*x                                            \   4    /                                        \   4    /                          |
                          ||- --------------------------------------------- + --------------------------------------------- + ---------------------------------------------   otherwise  |
                          ||       2        2/p*log(x)\    2    2/p*log(x)\        2        2/p*log(x)\    2    2/p*log(x)\        2        2/p*log(x)\    2    2/p*log(x)\              |
                          ||  4 + p  + 4*tan |--------| + p *tan |--------|   4 + p  + 4*tan |--------| + p *tan |--------|   4 + p  + 4*tan |--------| + p *tan |--------|              |
                          \\                 \   4    /          \   4    /                  \   4    /          \   4    /                  \   4    /          \   4    /              /
∫sin⁡(plog⁡(x)2) dx=C+{−ix24+ilog⁡(x)2for p=−2iix24−ilog⁡(x)2for p=2i2pxtan⁡2(plog⁡(x)4)p2tan⁡2(plog⁡(x)4)+p2+4tan⁡2(plog⁡(x)4)+4−2pxp2tan⁡2(plog⁡(x)4)+p2+4tan⁡2(plog⁡(x)4)+4+8xtan⁡(plog⁡(x)4)p2tan⁡2(plog⁡(x)4)+p2+4tan⁡2(plog⁡(x)4)+4otherwise\int \sin{\left(\frac{p \log{\left(x \right)}}{2} \right)}\, dx = C + \begin{cases} - \frac{i x^{2}}{4} + \frac{i \log{\left(x \right)}}{2} & \text{for}\: p = - 2 i \\\frac{i x^{2}}{4} - \frac{i \log{\left(x \right)}}{2} & \text{for}\: p = 2 i \\\frac{2 p x \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)}}{p^{2} \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + p^{2} + 4 \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + 4} - \frac{2 p x}{p^{2} \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + p^{2} + 4 \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + 4} + \frac{8 x \tan{\left(\frac{p \log{\left(x \right)}}{4} \right)}}{p^{2} \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + p^{2} + 4 \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + 4} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
        /p*log(pi)\             /p*log(pi)\
4*pi*sin|---------|   2*pi*p*cos|---------|
        \    2    /             \    2    /
------------------- - ---------------------
            2                      2       
       4 + p                  4 + p
−2πpcos⁡(plog⁡(π)2)p2+4+4πsin⁡(plog⁡(π)2)p2+4- \frac{2 \pi p \cos{\left(\frac{p \log{\left(\pi \right)}}{2} \right)}}{p^{2} + 4} + \frac{4 \pi \sin{\left(\frac{p \log{\left(\pi \right)}}{2} \right)}}{p^{2} + 4} 
=
= 
        /p*log(pi)\             /p*log(pi)\
4*pi*sin|---------|   2*pi*p*cos|---------|
        \    2    /             \    2    /
------------------- - ---------------------
            2                      2       
       4 + p                  4 + p
−2πpcos⁡(plog⁡(π)2)p2+4+4πsin⁡(plog⁡(π)2)p2+4- \frac{2 \pi p \cos{\left(\frac{p \log{\left(\pi \right)}}{2} \right)}}{p^{2} + 4} + \frac{4 \pi \sin{\left(\frac{p \log{\left(\pi \right)}}{2} \right)}}{p^{2} + 4} 
4*pi*sin(p*log(pi)/2)/(4 + p^2) - 2*pi*p*cos(p*log(pi)/2)/(4 + p^2)

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

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