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Resolución de integrales/ (sinx/sin(x-a))dx

Integral de (sinx/sin(x-a))dx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1              
  /              
 |               
 |    sin(x)     
 |  ---------- dx
 |  sin(x - a)   
 |               
/                
0
∫01sin⁡(x)sin⁡(−a+x) dx\int\limits_{0}^{1} \frac{\sin{\left(x \right)}}{\sin{\left(- a + x \right)}}\, dx 
Integral(sin(x)/sin(x - a), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                      /             
 |                      |              
 |   sin(x)             |   sin(x)     
 | ---------- dx = C +  | ---------- dx
 | sin(x - a)           | sin(x - a)   
 |                      |              
/                      /
∫sin⁡(x)sin⁡(−a+x) dx=C+∫sin⁡(x)sin⁡(−a+x) dx\int \frac{\sin{\left(x \right)}}{\sin{\left(- a + x \right)}}\, dx = C + \int \frac{\sin{\left(x \right)}}{\sin{\left(- a + x \right)}}\, dx
Simplificar
Respuesta [src] 
/                                 /  1   \    /a\                                                              /  1              \    /a\                                                                      
|                            2*log|------|*tan|-|                                                         2*log|------ + tan(1/2)|*tan|-|                                                                      
|                   2/a\          |   /a\|    \2/        /    /a\\    /a\        /       2     \    /a\        |   /a\           |    \2/        /     /a\           \    /a\                                  
|                tan |-|          |tan|-||          2*log|-tan|-||*tan|-|   2*log\1 + tan (1/2)/*tan|-|        |tan|-|           |          2*log|- tan|-| + tan(1/2)|*tan|-|                                  
|     1              \2/          \   \2//               \    \2//    \2/                           \2/        \   \2/           /               \     \2/           /    \2/                                  
<----------- - ----------- - -------------------- - --------------------- - --------------------------- + ------------------------------- + ---------------------------------  for And(a > -oo, a < oo, a != 0)
|       2/a\          2/a\              2/a\                    2/a\                       2/a\                            2/a\                               2/a\                                             
|1 + tan |-|   1 + tan |-|       1 + tan |-|             1 + tan |-|                1 + tan |-|                     1 + tan |-|                        1 + tan |-|                                             
|        \2/           \2/               \2/                     \2/                        \2/                             \2/                                \2/                                             
|                                                                                                                                                                                                              
\                                                                                     1                                                                                                   otherwise
{2log⁡(tan⁡(12)+1tan⁡(a2))tan⁡(a2)tan⁡2(a2)+1+2log⁡(−tan⁡(a2)+tan⁡(12))tan⁡(a2)tan⁡2(a2)+1−2log⁡(1tan⁡(a2))tan⁡(a2)tan⁡2(a2)+1−2log⁡(−tan⁡(a2))tan⁡(a2)tan⁡2(a2)+1−tan⁡2(a2)tan⁡2(a2)+1−2log⁡(tan⁡2(12)+1)tan⁡(a2)tan⁡2(a2)+1+1tan⁡2(a2)+1for a>−∞∧a<∞∧a≠01otherwise\begin{cases} \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \log{\left(- \tan{\left(\frac{a}{2} \right)} + \tan{\left(\frac{1}{2} \right)} \right)} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \log{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \log{\left(- \tan{\left(\frac{a}{2} \right)} \right)} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{\tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\1 & \text{otherwise} \end{cases} 
=
= 
/                                 /  1   \    /a\                                                              /  1              \    /a\                                                                      
|                            2*log|------|*tan|-|                                                         2*log|------ + tan(1/2)|*tan|-|                                                                      
|                   2/a\          |   /a\|    \2/        /    /a\\    /a\        /       2     \    /a\        |   /a\           |    \2/        /     /a\           \    /a\                                  
|                tan |-|          |tan|-||          2*log|-tan|-||*tan|-|   2*log\1 + tan (1/2)/*tan|-|        |tan|-|           |          2*log|- tan|-| + tan(1/2)|*tan|-|                                  
|     1              \2/          \   \2//               \    \2//    \2/                           \2/        \   \2/           /               \     \2/           /    \2/                                  
<----------- - ----------- - -------------------- - --------------------- - --------------------------- + ------------------------------- + ---------------------------------  for And(a > -oo, a < oo, a != 0)
|       2/a\          2/a\              2/a\                    2/a\                       2/a\                            2/a\                               2/a\                                             
|1 + tan |-|   1 + tan |-|       1 + tan |-|             1 + tan |-|                1 + tan |-|                     1 + tan |-|                        1 + tan |-|                                             
|        \2/           \2/               \2/                     \2/                        \2/                             \2/                                \2/                                             
|                                                                                                                                                                                                              
\                                                                                     1                                                                                                   otherwise
{2log⁡(tan⁡(12)+1tan⁡(a2))tan⁡(a2)tan⁡2(a2)+1+2log⁡(−tan⁡(a2)+tan⁡(12))tan⁡(a2)tan⁡2(a2)+1−2log⁡(1tan⁡(a2))tan⁡(a2)tan⁡2(a2)+1−2log⁡(−tan⁡(a2))tan⁡(a2)tan⁡2(a2)+1−tan⁡2(a2)tan⁡2(a2)+1−2log⁡(tan⁡2(12)+1)tan⁡(a2)tan⁡2(a2)+1+1tan⁡2(a2)+1for a>−∞∧a<∞∧a≠01otherwise\begin{cases} \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \log{\left(- \tan{\left(\frac{a}{2} \right)} + \tan{\left(\frac{1}{2} \right)} \right)} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \log{\left(\frac{1}{\tan{\left(\frac{a}{2} \right)}} \right)} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \log{\left(- \tan{\left(\frac{a}{2} \right)} \right)} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{\tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\1 & \text{otherwise} \end{cases} 
Piecewise((1/(1 + tan(a/2)^2) - tan(a/2)^2/(1 + tan(a/2)^2) - 2*log(1/tan(a/2))*tan(a/2)/(1 + tan(a/2)^2) - 2*log(-tan(a/2))*tan(a/2)/(1 + tan(a/2)^2) - 2*log(1 + tan(1/2)^2)*tan(a/2)/(1 + tan(a/2)^2) + 2*log(1/tan(a/2) + tan(1/2))*tan(a/2)/(1 + tan(a/2)^2) + 2*log(-tan(a/2) + tan(1/2))*tan(a/2)/(1 + tan(a/2)^2), (a > -oo)∧(a < oo)∧(Ne(a, 0))), (1, True))

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

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