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Resolución de integrales/ cos^2/ sin^2/ (2tgx+3)/(sin^2x+2cos^2x)

Integral de (2tgx+3)/(sin^2x+2cos^2x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                       
  /                       
 |                        
 |      2*tan(x) + 3      
 |  ------------------- dx
 |     2           2      
 |  sin (x) + 2*cos (x)   
 |                        
/                         
0
$$\int\limits_{0}^{1} \frac{2 \tan{\left(x \right)} + 3}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$$ 
Integral((2*tan(x) + 3)/(sin(x)^2 + 2*cos(x)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                                      /        /x   pi\                         \           /        /x   pi\                          \
                                                                      |        |- - --|                         |           |        |- - --|                          |
  /                                 /                             ___ |        |2   2 |       /      ___    /x\\|       ___ |        |2   2 |       /       ___    /x\\|
 |                                 |                          3*\/ 2 *|pi*floor|------| + atan|1 + \/ 2 *tan|-|||   3*\/ 2 *|pi*floor|------| + atan|-1 + \/ 2 *tan|-|||
 |     2*tan(x) + 3                |        tan(x)                    \        \  pi  /       \             \2///           \        \  pi  /       \              \2///
 | ------------------- dx = C + 2* | ------------------- dx + --------------------------------------------------- + ----------------------------------------------------
 |    2           2                |    2           2                                  2                                                     2                          
 | sin (x) + 2*cos (x)             | sin (x) + 2*cos (x)                                                                                                                
 |                                 |                                                                                                                                    
/                                 /
$$\int \frac{2 \tan{\left(x \right)} + 3}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx = C + \frac{3 \sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{3 \sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + 2 \int \frac{\tan{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$$
Simplificar
Respuesta [src] 
  1                       
  /                       
 |                        
 |      3 + 2*tan(x)      
 |  ------------------- dx
 |     2           2      
 |  sin (x) + 2*cos (x)   
 |                        
/                         
0
$$\int\limits_{0}^{1} \frac{2 \tan{\left(x \right)} + 3}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$$ 
=
= 
  1                       
  /                       
 |                        
 |      3 + 2*tan(x)      
 |  ------------------- dx
 |     2           2      
 |  sin (x) + 2*cos (x)   
 |                        
/                         
0
$$\int\limits_{0}^{1} \frac{2 \tan{\left(x \right)} + 3}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$$ 
Integral((3 + 2*tan(x))/(sin(x)^2 + 2*cos(x)^2), (x, 0, 1))
Respuesta numérica [src] 
2.56246295705989
2.56246295705989

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

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