Sr Examen

Otras calculadoras

Resolución de integrales/ cos(x)/ sin(x)/ (sin(x))/(1+sin(x)-cos(x))

Integral de (sin(x))/(1+sin(x)-cos(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                       
 --                       
 8                        
  /                       
 |                        
 |         sin(x)         
 |  ------------------- dx
 |  1 + sin(x) - cos(x)   
 |                        
/                         
pi                        
--                        
12
∫π12π8sin⁡(x)(sin⁡(x)+1)−cos⁡(x) dx\int\limits_{\frac{\pi}{12}}^{\frac{\pi}{8}} \frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) - \cos{\left(x \right)}}\, dx 
Integral(sin(x)/(1 + sin(x) - cos(x)), (x, pi/12, pi/8))
Solución detallada

  1. Vuelva a escribir el integrando:

    sin⁡(x)(sin⁡(x)+1)−cos⁡(x)=−sin⁡(x)−sin⁡(x)+cos⁡(x)−1\frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) - \cos{\left(x \right)}} = - \frac{\sin{\left(x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)} - 1}

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

    ∫(−sin⁡(x)−sin⁡(x)+cos⁡(x)−1) dx=−∫sin⁡(x)−sin⁡(x)+cos⁡(x)−1 dx\int \left(- \frac{\sin{\left(x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)} - 1}\right)\, dx = - \int \frac{\sin{\left(x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)} - 1}\, dx

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

      Pero la integral

      −x2−log⁡(tan⁡(x2)+1)+log⁡(tan⁡2(x2)+1)2- \frac{x}{2} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{2}

    Por lo tanto, el resultado es: x2+log⁡(tan⁡(x2)+1)−log⁡(tan⁡2(x2)+1)2\frac{x}{2} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{2}

  3. Ahora simplificar:

    x2−log⁡(2cos⁡(x)+1)2+log⁡(tan⁡(x2)+1)\frac{x}{2} - \frac{\log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)}}{2} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}

  4. Añadimos la constante de integración:

    x2−log⁡(2cos⁡(x)+1)2+log⁡(tan⁡(x2)+1)+constant\frac{x}{2} - \frac{\log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)}}{2} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}+ \mathrm{constant}

Respuesta:

x2−log⁡(2cos⁡(x)+1)2+log⁡(tan⁡(x2)+1)+constant\frac{x}{2} - \frac{\log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)}}{2} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                    /       2/x\\                  
 |                                  log|1 + tan |-||                  
 |        sin(x)                x      \        \2//      /       /x\\
 | ------------------- dx = C + - - ---------------- + log|1 + tan|-||
 | 1 + sin(x) - cos(x)          2          2              \       \2//
 |                                                                    
/
∫sin⁡(x)(sin⁡(x)+1)−cos⁡(x) dx=C+x2+log⁡(tan⁡(x2)+1)−log⁡(tan⁡2(x2)+1)2\int \frac{\sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) - \cos{\left(x \right)}}\, dx = C + \frac{x}{2} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{2}
Simplificar
Gráfica
0.270.280.290.300.310.320.330.340.350.360.370.380.390.01.0
Trazar el gráfico f(x)
Respuesta [src] 
                                                                                                /                                         2\                                                    
                                                                                                |    /                        ___________\ |                                                    
   /                                       2\                                                   |    |                       /       ___ | |                                                    
   |    /                     ___     ___ \ |                                                   |    |                      /  1   \/ 2  | |                                                    
   |    |                   \/ 2    \/ 6  | |                                                   |    |                     /   - + ----- | |                                                    
   |    |                   ----- + ----- | |                                                   |    |       1           \/    2     4   | |                                                    
   |    |       1             4       4   | |                                                log|1 + |---------------- - ----------------| |                                                    
log|1 + |--------------- - ---------------| |                                                   |    |     ___________        ___________| |           /                            ___________\
   |    |    ___     ___       ___     ___| |      /                         ___     ___ \      |    |    /       ___        /       ___ | |           |                           /       ___ |
   |    |  \/ 2    \/ 6      \/ 2    \/ 6 | |      |                       \/ 2    \/ 6  |      |    |   /  1   \/ 2        /  1   \/ 2  | |           |                          /  1   \/ 2  |
   |    |- ----- + -----   - ----- + -----| |      |                       ----- + ----- |      |    |  /   - - -----      /   - - ----- | |           |                         /   - + ----- |
   \    \    4       4         4       4  / /      |           1             4       4   |      \    \\/    2     4      \/    2     4   / /   pi      |           1           \/    2     4   |
--------------------------------------------- - log|1 + --------------- - ---------------| - ----------------------------------------------- + -- + log|1 + ---------------- - ----------------|
                      2                            |        ___     ___       ___     ___|                          2                          48      |         ___________        ___________|
                                                   |      \/ 2    \/ 6      \/ 2    \/ 6 |                                                             |        /       ___        /       ___ |
                                                   |    - ----- + -----   - ----- + -----|                                                             |       /  1   \/ 2        /  1   \/ 2  |
                                                   \        4       4         4       4  /                                                             |      /   - - -----      /   - - ----- |
                                                                                                                                                       \    \/    2     4      \/    2     4   /
−log⁡(−24+64−24+64+1+1−24+64)−log⁡((−24+1212−24+112−24)2+1)2+log⁡((−24+64−24+64+1−24+64)2+1)2+π48+log⁡(−24+1212−24+1+112−24)- \log{\left(- \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + 1 + \frac{1}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} \right)} - \frac{\log{\left(\left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} + \frac{1}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}\right)^{2} + 1 \right)}}{2} + \frac{\log{\left(\left(- \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + \frac{1}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}\right)^{2} + 1 \right)}}{2} + \frac{\pi}{48} + \log{\left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} + 1 + \frac{1}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} \right)} 
=
= 
                                                                                                /                                         2\                                                    
                                                                                                |    /                        ___________\ |                                                    
   /                                       2\                                                   |    |                       /       ___ | |                                                    
   |    /                     ___     ___ \ |                                                   |    |                      /  1   \/ 2  | |                                                    
   |    |                   \/ 2    \/ 6  | |                                                   |    |                     /   - + ----- | |                                                    
   |    |                   ----- + ----- | |                                                   |    |       1           \/    2     4   | |                                                    
   |    |       1             4       4   | |                                                log|1 + |---------------- - ----------------| |                                                    
log|1 + |--------------- - ---------------| |                                                   |    |     ___________        ___________| |           /                            ___________\
   |    |    ___     ___       ___     ___| |      /                         ___     ___ \      |    |    /       ___        /       ___ | |           |                           /       ___ |
   |    |  \/ 2    \/ 6      \/ 2    \/ 6 | |      |                       \/ 2    \/ 6  |      |    |   /  1   \/ 2        /  1   \/ 2  | |           |                          /  1   \/ 2  |
   |    |- ----- + -----   - ----- + -----| |      |                       ----- + ----- |      |    |  /   - - -----      /   - - ----- | |           |                         /   - + ----- |
   \    \    4       4         4       4  / /      |           1             4       4   |      \    \\/    2     4      \/    2     4   / /   pi      |           1           \/    2     4   |
--------------------------------------------- - log|1 + --------------- - ---------------| - ----------------------------------------------- + -- + log|1 + ---------------- - ----------------|
                      2                            |        ___     ___       ___     ___|                          2                          48      |         ___________        ___________|
                                                   |      \/ 2    \/ 6      \/ 2    \/ 6 |                                                             |        /       ___        /       ___ |
                                                   |    - ----- + -----   - ----- + -----|                                                             |       /  1   \/ 2        /  1   \/ 2  |
                                                   \        4       4         4       4  /                                                             |      /   - - -----      /   - - ----- |
                                                                                                                                                       \    \/    2     4      \/    2     4   /
−log⁡(−24+64−24+64+1+1−24+64)−log⁡((−24+1212−24+112−24)2+1)2+log⁡((−24+64−24+64+1−24+64)2+1)2+π48+log⁡(−24+1212−24+1+112−24)- \log{\left(- \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + 1 + \frac{1}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} \right)} - \frac{\log{\left(\left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} + \frac{1}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}\right)^{2} + 1 \right)}}{2} + \frac{\log{\left(\left(- \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + \frac{1}{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}\right)^{2} + 1 \right)}}{2} + \frac{\pi}{48} + \log{\left(- \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} + 1 + \frac{1}{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}} \right)} 
log(1 + (1/(-sqrt(2)/4 + sqrt(6)/4) - (sqrt(2)/4 + sqrt(6)/4)/(-sqrt(2)/4 + sqrt(6)/4))^2)/2 - log(1 + 1/(-sqrt(2)/4 + sqrt(6)/4) - (sqrt(2)/4 + sqrt(6)/4)/(-sqrt(2)/4 + sqrt(6)/4)) - log(1 + (1/sqrt(1/2 - sqrt(2)/4) - sqrt(1/2 + sqrt(2)/4)/sqrt(1/2 - sqrt(2)/4))^2)/2 + pi/48 + log(1 + 1/sqrt(1/2 - sqrt(2)/4) - sqrt(1/2 + sqrt(2)/4)/sqrt(1/2 - sqrt(2)/4))
Respuesta numérica [src] 
0.112375902743695
0.112375902743695

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

    © Sr Examen – Калькулятор