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Resolución de integrales/ sin(x)/ (sinx)/ ln(2/(sinx))*sin(x)

Integral de ln(2/(sinx))*sin(x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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∫01log⁡(2sin⁡(x))sin⁡(x) dx\int\limits_{0}^{1} \log{\left(\frac{2}{\sin{\left(x \right)}} \right)} \sin{\left(x \right)}\, dx 
Integral(log(2/sin(x))*sin(x), (x, 0, 1))
Solución detallada

  1. Hay varias maneras de calcular esta integral.

    Método #1

    1. Vuelva a escribir el integrando:

      log⁡(2sin⁡(x))sin⁡(x)=log⁡(1sin⁡(x))sin⁡(x)+log⁡(2)sin⁡(x)\log{\left(\frac{2}{\sin{\left(x \right)}} \right)} \sin{\left(x \right)} = \log{\left(\frac{1}{\sin{\left(x \right)}} \right)} \sin{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)}

    2. Integramos término a término:

      1. Usamos la integración por partes:

        ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

        que u(x)=log⁡(1sin⁡(x))u{\left(x \right)} = \log{\left(\frac{1}{\sin{\left(x \right)}} \right)} y que dv⁡(x)=sin⁡(x)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

        Entonces du⁡(x)=−cos⁡(x)sin⁡(x)\operatorname{du}{\left(x \right)} = - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}.

        Para buscar v(x)v{\left(x \right)}:

        1. La integral del seno es un coseno menos:

          ∫sin⁡(x) dx=−cos⁡(x)\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}

        Ahora resolvemos podintegral.

      2. No puedo encontrar los pasos en la búsqueda de esta integral.

        Pero la integral

        log⁡(cos⁡(x)−1)2−log⁡(cos⁡(x)+1)2+cos⁡(x)\frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2} + \cos{\left(x \right)}

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫log⁡(2)sin⁡(x) dx=log⁡(2)∫sin⁡(x) dx\int \log{\left(2 \right)} \sin{\left(x \right)}\, dx = \log{\left(2 \right)} \int \sin{\left(x \right)}\, dx

        1. La integral del seno es un coseno menos:

          ∫sin⁡(x) dx=−cos⁡(x)\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}

        Por lo tanto, el resultado es: −log⁡(2)cos⁡(x)- \log{\left(2 \right)} \cos{\left(x \right)}

      El resultado es: −log⁡(cos⁡(x)−1)2+log⁡(cos⁡(x)+1)2−log⁡(1sin⁡(x))cos⁡(x)−cos⁡(x)−log⁡(2)cos⁡(x)- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2} - \log{\left(\frac{1}{\sin{\left(x \right)}} \right)} \cos{\left(x \right)} - \cos{\left(x \right)} - \log{\left(2 \right)} \cos{\left(x \right)}

    Método #2

    1. Usamos la integración por partes:

      ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

      que u(x)=log⁡(2sin⁡(x))u{\left(x \right)} = \log{\left(\frac{2}{\sin{\left(x \right)}} \right)} y que dv⁡(x)=sin⁡(x)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

      Entonces du⁡(x)=−cos⁡(x)sin⁡(x)\operatorname{du}{\left(x \right)} = - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}.

      Para buscar v(x)v{\left(x \right)}:

      1. La integral del seno es un coseno menos:

        ∫sin⁡(x) dx=−cos⁡(x)\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}

      Ahora resolvemos podintegral.

    2. No puedo encontrar los pasos en la búsqueda de esta integral.

      Pero la integral

      log⁡(cos⁡(x)−1)2−log⁡(cos⁡(x)+1)2+cos⁡(x)\frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2} + \cos{\left(x \right)}

  2. Añadimos la constante de integración:

    −log⁡(cos⁡(x)−1)2+log⁡(cos⁡(x)+1)2−log⁡(1sin⁡(x))cos⁡(x)−cos⁡(x)−log⁡(2)cos⁡(x)+constant- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2} - \log{\left(\frac{1}{\sin{\left(x \right)}} \right)} \cos{\left(x \right)} - \cos{\left(x \right)} - \log{\left(2 \right)} \cos{\left(x \right)}+ \mathrm{constant}

Respuesta:

−log⁡(cos⁡(x)−1)2+log⁡(cos⁡(x)+1)2−log⁡(1sin⁡(x))cos⁡(x)−cos⁡(x)−log⁡(2)cos⁡(x)+constant- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2} - \log{\left(\frac{1}{\sin{\left(x \right)}} \right)} \cos{\left(x \right)} - \cos{\left(x \right)} - \log{\left(2 \right)} \cos{\left(x \right)}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                                            
 |                                                                                                             
 |    /  2   \                 log(1 + cos(x))            log(-1 + cos(x))                             /  1   \
 | log|------|*sin(x) dx = C + --------------- - cos(x) - ---------------- - cos(x)*log(2) - cos(x)*log|------|
 |    \sin(x)/                        2                          2                                     \sin(x)/
 |                                                                                                             
/
∫log⁡(2sin⁡(x))sin⁡(x) dx=C−log⁡(cos⁡(x)−1)2+log⁡(cos⁡(x)+1)2−log⁡(1sin⁡(x))cos⁡(x)−cos⁡(x)−log⁡(2)cos⁡(x)\int \log{\left(\frac{2}{\sin{\left(x \right)}} \right)} \sin{\left(x \right)}\, dx = C - \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2} - \log{\left(\frac{1}{\sin{\left(x \right)}} \right)} \cos{\left(x \right)} - \cos{\left(x \right)} - \log{\left(2 \right)} \cos{\left(x \right)}
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Gráfica
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Trazar el gráfico f(x)
Respuesta [src] 
                                                                             2         /   1               \
                       /       2     \      2         /       2     \   2*tan (1/2)*log|-------- + tan(1/2)|
          2         log\1 + tan (1/2)/   tan (1/2)*log\1 + tan (1/2)/                  \tan(1/2)           /
2 - ------------- - ------------------ - ---------------------------- + ------------------------------------
           2                 2                         2                                  2                 
    1 + tan (1/2)     1 + tan (1/2)             1 + tan (1/2)                      1 + tan (1/2)
−2tan⁡2(12)+1−log⁡(tan⁡2(12)+1)tan⁡2(12)+1−log⁡(tan⁡2(12)+1)tan⁡2(12)tan⁡2(12)+1+2log⁡(tan⁡(12)+1tan⁡(12))tan⁡2(12)tan⁡2(12)+1+2- \frac{2}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{1}{\tan{\left(\frac{1}{2} \right)}} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + 2 
=
= 
                                                                             2         /   1               \
                       /       2     \      2         /       2     \   2*tan (1/2)*log|-------- + tan(1/2)|
          2         log\1 + tan (1/2)/   tan (1/2)*log\1 + tan (1/2)/                  \tan(1/2)           /
2 - ------------- - ------------------ - ---------------------------- + ------------------------------------
           2                 2                         2                                  2                 
    1 + tan (1/2)     1 + tan (1/2)             1 + tan (1/2)                      1 + tan (1/2)
−2tan⁡2(12)+1−log⁡(tan⁡2(12)+1)tan⁡2(12)+1−log⁡(tan⁡2(12)+1)tan⁡2(12)tan⁡2(12)+1+2log⁡(tan⁡(12)+1tan⁡(12))tan⁡2(12)tan⁡2(12)+1+2- \frac{2}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{1}{\tan{\left(\frac{1}{2} \right)}} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + 2 
2 - 2/(1 + tan(1/2)^2) - log(1 + tan(1/2)^2)/(1 + tan(1/2)^2) - tan(1/2)^2*log(1 + tan(1/2)^2)/(1 + tan(1/2)^2) + 2*tan(1/2)^2*log(1/tan(1/2) + tan(1/2))/(1 + tan(1/2)^2)
Respuesta numérica [src] 
0.596512918000244
0.596512918000244

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

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