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Resolución de integrales/ dx/cos/ cos³x/ sincdx/cos³x

Integral de sincdx/cos³x dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1           
  /           
 |            
 |  sinc(1)   
 |  ------- dx
 |     3      
 |  cos (x)   
 |            
/             
0
∫01sinc⁡(1)cos⁡3(x) dx\int\limits_{0}^{1} \frac{\operatorname{sinc}{\left(1 \right)}}{\cos^{3}{\left(x \right)}}\, dx 
Integral(sinc(1)/cos(x)^3, (x, 0, 1))
Solución detallada

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

    ∫sinc⁡(1)cos⁡3(x) dx=sinc⁡(1)∫1cos⁡3(x) dx\int \frac{\operatorname{sinc}{\left(1 \right)}}{\cos^{3}{\left(x \right)}}\, dx = \operatorname{sinc}{\left(1 \right)} \int \frac{1}{\cos^{3}{\left(x \right)}}\, dx

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

      Pero la integral

      −log⁡(sin⁡(x)−1)4+log⁡(sin⁡(x)+1)4−sin⁡(x)2sin⁡2(x)−2- \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{4} - \frac{\sin{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - 2}

    Por lo tanto, el resultado es: (−log⁡(sin⁡(x)−1)4+log⁡(sin⁡(x)+1)4−sin⁡(x)2sin⁡2(x)−2)sinc⁡(1)\left(- \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{4} - \frac{\sin{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - 2}\right) \operatorname{sinc}{\left(1 \right)}

  2. Ahora simplificar:

    (−log⁡(sin⁡(x)−1)+log⁡(sin⁡(x)+1)+2sin⁡(x)cos⁡2(x))sinc⁡(1)4\frac{\left(- \log{\left(\sin{\left(x \right)} - 1 \right)} + \log{\left(\sin{\left(x \right)} + 1 \right)} + \frac{2 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \operatorname{sinc}{\left(1 \right)}}{4}

  3. Añadimos la constante de integración:

    (−log⁡(sin⁡(x)−1)+log⁡(sin⁡(x)+1)+2sin⁡(x)cos⁡2(x))sinc⁡(1)4+constant\frac{\left(- \log{\left(\sin{\left(x \right)} - 1 \right)} + \log{\left(\sin{\left(x \right)} + 1 \right)} + \frac{2 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \operatorname{sinc}{\left(1 \right)}}{4}+ \mathrm{constant}

Respuesta:

(−log⁡(sin⁡(x)−1)+log⁡(sin⁡(x)+1)+2sin⁡(x)cos⁡2(x))sinc⁡(1)4+constant\frac{\left(- \log{\left(\sin{\left(x \right)} - 1 \right)} + \log{\left(\sin{\left(x \right)} + 1 \right)} + \frac{2 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \operatorname{sinc}{\left(1 \right)}}{4}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                
 |                                                                                 
 | sinc(1)          /  log(-1 + sin(x))   log(1 + sin(x))       sin(x)    \        
 | ------- dx = C + |- ---------------- + --------------- - --------------|*sinc(1)
 |    3             |         4                  4                    2   |        
 | cos (x)          \                                       -2 + 2*sin (x)/        
 |                                                                                 
/
∫sinc⁡(1)cos⁡3(x) dx=C+(−log⁡(sin⁡(x)−1)4+log⁡(sin⁡(x)+1)4−sin⁡(x)2sin⁡2(x)−2)sinc⁡(1)\int \frac{\operatorname{sinc}{\left(1 \right)}}{\cos^{3}{\left(x \right)}}\, dx = C + \left(- \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{4} - \frac{\sin{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - 2}\right) \operatorname{sinc}{\left(1 \right)}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90010
Trazar el gráfico f(x)
Respuesta [src] 
/  log(1 - sin(1))   log(1 + sin(1))       sin(1)       pi*I\           pi*I*sinc(1)
|- --------------- + --------------- - -------------- - ----|*sinc(1) + ------------
|         4                 4                    2       4  |                4      
\                                      -2 + 2*sin (1)       /
(log⁡(sin⁡(1)+1)4−log⁡(1−sin⁡(1))4−sin⁡(1)−2+2sin⁡2(1)−iπ4)sinc⁡(1)+iπsinc⁡(1)4\left(\frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{4} - \frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{4} - \frac{\sin{\left(1 \right)}}{-2 + 2 \sin^{2}{\left(1 \right)}} - \frac{i \pi}{4}\right) \operatorname{sinc}{\left(1 \right)} + \frac{i \pi \operatorname{sinc}{\left(1 \right)}}{4} 
=
= 
/  log(1 - sin(1))   log(1 + sin(1))       sin(1)       pi*I\           pi*I*sinc(1)
|- --------------- + --------------- - -------------- - ----|*sinc(1) + ------------
|         4                 4                    2       4  |                4      
\                                      -2 + 2*sin (1)       /
(log⁡(sin⁡(1)+1)4−log⁡(1−sin⁡(1))4−sin⁡(1)−2+2sin⁡2(1)−iπ4)sinc⁡(1)+iπsinc⁡(1)4\left(\frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{4} - \frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{4} - \frac{\sin{\left(1 \right)}}{-2 + 2 \sin^{2}{\left(1 \right)}} - \frac{i \pi}{4}\right) \operatorname{sinc}{\left(1 \right)} + \frac{i \pi \operatorname{sinc}{\left(1 \right)}}{4} 
(-log(1 - sin(1))/4 + log(1 + sin(1))/4 - sin(1)/(-2 + 2*sin(1)^2) - pi*i/4)*sinc(1) + pi*i*sinc(1)/4
Respuesta numérica [src] 
1.72866155647043
1.72866155647043

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

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