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Resolución de integrales/ cos^2/ sin^3/ (sin^3xdx)/(1+cos^2x)

Integral de (sin^3xdx)/(1+cos^2x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |       3        
 |    sin (x)     
 |  ----------- dx
 |         2      
 |  1 + cos (x)   
 |                
/                 
0
$$\int\limits_{0}^{1} \frac{\sin^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 1}\, dx$$ 
Integral(sin(x)^3/(1 + cos(x)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                        /        /x   pi\                         \     /        /x   pi\                          \             /        /x   pi\                         \             /        /x   pi\                          \
  /                                     |        |- - --|                         |     |        |- - --|                          |             |        |- - --|                         |             |        |- - --|                          |
 |                                      |        |2   2 |       /      ___    /x\\|     |        |2   2 |       /       ___    /x\\|        2/x\ |        |2   2 |       /      ___    /x\\|        2/x\ |        |2   2 |       /       ___    /x\\|
 |      3                             2*|pi*floor|------| + atan|1 + \/ 2 *tan|-|||   2*|pi*floor|------| + atan|-1 + \/ 2 *tan|-|||   2*tan |-|*|pi*floor|------| + atan|1 + \/ 2 *tan|-|||   2*tan |-|*|pi*floor|------| + atan|-1 + \/ 2 *tan|-|||
 |   sin (x)                 2          \        \  pi  /       \             \2///     \        \  pi  /       \              \2///         \2/ \        \  pi  /       \             \2///         \2/ \        \  pi  /       \              \2///
 | ----------- dx = C + ----------- - --------------------------------------------- + ---------------------------------------------- - ----------------------------------------------------- + ------------------------------------------------------
 |        2                    2/x\                           2/x\                                            2/x\                                                 2/x\                                                    2/x\                      
 | 1 + cos (x)          1 + tan |-|                    1 + tan |-|                                     1 + tan |-|                                          1 + tan |-|                                             1 + tan |-|                      
 |                              \2/                            \2/                                             \2/                                                  \2/                                                     \2/                      
/
$$\int \frac{\sin^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 1}\, dx = C + \frac{2 \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
Simplificar
Respuesta [src] 
                            /          /      ___         \\     /          /      ___         \\        2      /          /      ___         \\        2      /          /      ___         \\
                2         2*\-pi + atan\1 + \/ 2 *tan(1/2)//   2*\-pi - atan\1 - \/ 2 *tan(1/2)//   2*tan (1/2)*\-pi + atan\1 + \/ 2 *tan(1/2)//   2*tan (1/2)*\-pi - atan\1 - \/ 2 *tan(1/2)//
-2 + pi + ------------- - ---------------------------------- + ---------------------------------- - -------------------------------------------- + --------------------------------------------
                 2                         2                                    2                                         2                                              2                     
          1 + tan (1/2)             1 + tan (1/2)                        1 + tan (1/2)                             1 + tan (1/2)                                  1 + tan (1/2)
$$\frac{2 \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - 2 + \frac{2 \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \pi - \frac{2 \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{\tan^{2}{\left(\frac{1}{2} \right)} + 1}$$ 
=
= 
                            /          /      ___         \\     /          /      ___         \\        2      /          /      ___         \\        2      /          /      ___         \\
                2         2*\-pi + atan\1 + \/ 2 *tan(1/2)//   2*\-pi - atan\1 - \/ 2 *tan(1/2)//   2*tan (1/2)*\-pi + atan\1 + \/ 2 *tan(1/2)//   2*tan (1/2)*\-pi - atan\1 - \/ 2 *tan(1/2)//
-2 + pi + ------------- - ---------------------------------- + ---------------------------------- - -------------------------------------------- + --------------------------------------------
                 2                         2                                    2                                         2                                              2                     
          1 + tan (1/2)             1 + tan (1/2)                        1 + tan (1/2)                             1 + tan (1/2)                                  1 + tan (1/2)
$$\frac{2 \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - 2 + \frac{2 \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \pi - \frac{2 \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{\tan^{2}{\left(\frac{1}{2} \right)} + 1}$$ 
-2 + pi + 2/(1 + tan(1/2)^2) - 2*(-pi + atan(1 + sqrt(2)*tan(1/2)))/(1 + tan(1/2)^2) + 2*(-pi - atan(1 - sqrt(2)*tan(1/2)))/(1 + tan(1/2)^2) - 2*tan(1/2)^2*(-pi + atan(1 + sqrt(2)*tan(1/2)))/(1 + tan(1/2)^2) + 2*tan(1/2)^2*(-pi - atan(1 - sqrt(2)*tan(1/2)))/(1 + tan(1/2)^2)
Respuesta numérica [src] 
0.12036405422569
0.12036405422569

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

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