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Resolución de integrales/ sin(x)/ x/x^2/ (1+sin(x))dx/x^2

Integral de (1+sin(x))dx/x^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo              
  /              
 |               
 |  1 + sin(x)   
 |  ---------- dx
 |       2       
 |      x        
 |               
/                
2
∫2∞sin⁡(x)+1x2 dx\int\limits_{2}^{\infty} \frac{\sin{\left(x \right)} + 1}{x^{2}}\, dx 
Integral((1 + sin(x))/x^2, (x, 2, oo))
Solución detallada

  1. Vuelva a escribir el integrando:

    sin⁡(x)+1x2=sin⁡(x)x2+1x2\frac{\sin{\left(x \right)} + 1}{x^{2}} = \frac{\sin{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}}

  2. Integramos término a término:

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

      Pero la integral

      −log⁡(x)+log⁡(x2)2+Ci⁡(x)−sin⁡(x)x- \log{\left(x \right)} + \frac{\log{\left(x^{2} \right)}}{2} + \operatorname{Ci}{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}

    1. Integral xnx^{n} es xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫1x2 dx=−1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

    El resultado es: −log⁡(x)+log⁡(x2)2+Ci⁡(x)−sin⁡(x)x−1x- \log{\left(x \right)} + \frac{\log{\left(x^{2} \right)}}{2} + \operatorname{Ci}{\left(x \right)} - \frac{\sin{\left(x \right)}}{x} - \frac{1}{x}

  3. Añadimos la constante de integración:

    −log⁡(x)+log⁡(x2)2+Ci⁡(x)−sin⁡(x)x−1x+constant- \log{\left(x \right)} + \frac{\log{\left(x^{2} \right)}}{2} + \operatorname{Ci}{\left(x \right)} - \frac{\sin{\left(x \right)}}{x} - \frac{1}{x}+ \mathrm{constant}

Respuesta:

−log⁡(x)+log⁡(x2)2+Ci⁡(x)−sin⁡(x)x−1x+constant- \log{\left(x \right)} + \frac{\log{\left(x^{2} \right)}}{2} + \operatorname{Ci}{\left(x \right)} - \frac{\sin{\left(x \right)}}{x} - \frac{1}{x}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                         
 |                        / 2\                              
 | 1 + sin(x)          log\x /   1            sin(x)        
 | ---------- dx = C + ------- - - - log(x) - ------ + Ci(x)
 |      2                 2      x              x           
 |     x                                                    
 |                                                          
/
∫sin⁡(x)+1x2 dx=C−log⁡(x)+log⁡(x2)2+Ci⁡(x)−sin⁡(x)x−1x\int \frac{\sin{\left(x \right)} + 1}{x^{2}}\, dx = C - \log{\left(x \right)} + \frac{\log{\left(x^{2} \right)}}{2} + \operatorname{Ci}{\left(x \right)} - \frac{\sin{\left(x \right)}}{x} - \frac{1}{x}
Simplificar
Respuesta [src] 
           |                                     /                                   /   pi*I\                    \|
           |                                     |                                   |   ----|                    ||
           |                                     |                                   |    2  |   3*sin(2)   3*pi*I||
           |                                   4*|-3 - 3*Ci(2) + 3*EulerGamma + 3*log\2*e    / + -------- - ------||
      ____ |  4      4*EulerGamma   4*log(2)     \                                                  2         2   /|
    \/ pi *|------ - ------------ - -------- + --------------------------------------------------------------------|
           |  ____        ____         ____                                      ____                              |
1          |\/ pi       \/ pi        \/ pi                                   3*\/ pi                               |
- + ----------------------------------------------------------------------------------------------------------------
2                                                          4
π∣−4log⁡(2)π−4γπ+4π+4(−3−3Ci⁡(2)+3sin⁡(2)2+3γ−3iπ2+3log⁡(2eiπ2))3π∣4+12\frac{\sqrt{\pi} \left|{- \frac{4 \log{\left(2 \right)}}{\sqrt{\pi}} - \frac{4 \gamma}{\sqrt{\pi}} + \frac{4}{\sqrt{\pi}} + \frac{4 \left(-3 - 3 \operatorname{Ci}{\left(2 \right)} + \frac{3 \sin{\left(2 \right)}}{2} + 3 \gamma - \frac{3 i \pi}{2} + 3 \log{\left(2 e^{\frac{i \pi}{2}} \right)}\right)}{3 \sqrt{\pi}}}\right|}{4} + \frac{1}{2} 
=
= 
           |                                     /                                   /   pi*I\                    \|
           |                                     |                                   |   ----|                    ||
           |                                     |                                   |    2  |   3*sin(2)   3*pi*I||
           |                                   4*|-3 - 3*Ci(2) + 3*EulerGamma + 3*log\2*e    / + -------- - ------||
      ____ |  4      4*EulerGamma   4*log(2)     \                                                  2         2   /|
    \/ pi *|------ - ------------ - -------- + --------------------------------------------------------------------|
           |  ____        ____         ____                                      ____                              |
1          |\/ pi       \/ pi        \/ pi                                   3*\/ pi                               |
- + ----------------------------------------------------------------------------------------------------------------
2                                                          4
π∣−4log⁡(2)π−4γπ+4π+4(−3−3Ci⁡(2)+3sin⁡(2)2+3γ−3iπ2+3log⁡(2eiπ2))3π∣4+12\frac{\sqrt{\pi} \left|{- \frac{4 \log{\left(2 \right)}}{\sqrt{\pi}} - \frac{4 \gamma}{\sqrt{\pi}} + \frac{4}{\sqrt{\pi}} + \frac{4 \left(-3 - 3 \operatorname{Ci}{\left(2 \right)} + \frac{3 \sin{\left(2 \right)}}{2} + 3 \gamma - \frac{3 i \pi}{2} + 3 \log{\left(2 e^{\frac{i \pi}{2}} \right)}\right)}{3 \sqrt{\pi}}}\right|}{4} + \frac{1}{2} 
1/2 + sqrt(pi)*Abs(4/sqrt(pi) - 4*EulerGamma/sqrt(pi) - 4*log(2)/sqrt(pi) + 4*(-3 - 3*Ci(2) + 3*EulerGamma + 3*log(2*exp_polar(pi*i/2)) + 3*sin(2)/2 - 3*pi*i/2)/(3*sqrt(pi)))/4

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

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