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Resolución de integrales/ sin(x)/ 1/(1+2sin(x)*sin(x))

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                       
 --                       
 4                        
  /                       
 |                        
 |           1            
 |  ------------------- dx
 |  1 + 2*sin(x)*sin(x)   
 |                        
/                         
0
$$\int\limits_{0}^{\frac{\pi}{4}} \frac{1}{\sin{\left(x \right)} 2 \sin{\left(x \right)} + 1}\, dx$$ 
Integral(1/(1 + (2*sin(x))*sin(x)), (x, 0, pi/4))
Respuesta (Indefinida) [src] 
                                   /        /x   pi\       /        /x\     \\            /        /x   pi\       /        /x\     \\                                       /        /x   pi\       /        /x\     \\                                             /        /x   pi\       /        /x\     \\
                                   |        |- - --|       |     tan|-|     ||            |        |- - --|       |     tan|-|     ||        _____________    _____________ |        |- - --|       |     tan|-|     ||              _____________    _____________ |        |- - --|       |     tan|-|     ||
                                   |        |2   2 |       |        \2/     ||        ___ |        |2   2 |       |        \2/     ||       /         ___    /         ___  |        |2   2 |       |        \2/     ||       ___   /         ___    /         ___  |        |2   2 |       |        \2/     ||
                                 5*|pi*floor|------| + atan|----------------||    2*\/ 6 *|pi*floor|------| + atan|----------------||   5*\/  5 - 2*\/ 6  *\/  5 + 2*\/ 6  *|pi*floor|------| + atan|----------------||   2*\/ 6 *\/  5 - 2*\/ 6  *\/  5 + 2*\/ 6  *|pi*floor|------| + atan|----------------||
  /                                |        \  pi  /       |   _____________||            |        \  pi  /       |   _____________||                                       |        \  pi  /       |   _____________||                                             |        \  pi  /       |   _____________||
 |                                 |                       |  /         ___ ||            |                       |  /         ___ ||                                       |                       |  /         ___ ||                                             |                       |  /         ___ ||
 |          1                      \                       \\/  5 - 2*\/ 6  //            \                       \\/  5 - 2*\/ 6  //                                       \                       \\/  5 + 2*\/ 6  //                                             \                       \\/  5 + 2*\/ 6  //
 | ------------------- dx = C + ----------------------------------------------- + --------------------------------------------------- + ------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
 | 1 + 2*sin(x)*sin(x)                _____________               _____________           _____________               _____________                           _____________               _____________                                            _____________               _____________                   
 |                                   /         ___         ___   /         ___           /         ___         ___   /         ___                           /         ___         ___   /         ___                                            /         ___         ___   /         ___                    
/                               27*\/  5 - 2*\/ 6   + 11*\/ 6 *\/  5 - 2*\/ 6       27*\/  5 - 2*\/ 6   + 11*\/ 6 *\/  5 - 2*\/ 6                       27*\/  5 - 2*\/ 6   + 11*\/ 6 *\/  5 - 2*\/ 6                                        27*\/  5 - 2*\/ 6   + 11*\/ 6 *\/  5 - 2*\/ 6
$$\int \frac{1}{\sin{\left(x \right)} 2 \sin{\left(x \right)} + 1}\, dx = C + \frac{2 \sqrt{6} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{5 - 2 \sqrt{6}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{5 \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{5 - 2 \sqrt{6}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{2 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{6} + 5}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}} + \frac{5 \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{6} + 5}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{11 \sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 27 \sqrt{5 - 2 \sqrt{6}}}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                                                /          ___   \                                                           _____________    _____________ /          /          ___   \\
                                                                |   -1 + \/ 2    |                                                          /         ___    /         ___  |          |   -1 + \/ 2    ||
                                                      -pi + atan|----------------|                                                        \/  5 - 2*\/ 6  *\/  5 + 2*\/ 6  *|-pi + atan|----------------||
                                                                |   _____________|                    _____________    _____________                                        |          |   _____________||
                                                                |  /         ___ |                   /         ___    /         ___                                         |          |  /         ___ ||
                     pi                                         \\/  5 - 2*\/ 6  /              pi*\/  5 - 2*\/ 6  *\/  5 + 2*\/ 6                                          \          \\/  5 + 2*\/ 6  //
------------------------------------------- + ------------------------------------------- + ------------------------------------------- + ----------------------------------------------------------------
     _____________            _____________        _____________            _____________        _____________            _____________                  _____________            _____________           
    /         ___      ___   /         ___        /         ___      ___   /         ___        /         ___      ___   /         ___                  /         ___      ___   /         ___            
3*\/  5 - 2*\/ 6   + \/ 6 *\/  5 - 2*\/ 6     3*\/  5 - 2*\/ 6   + \/ 6 *\/  5 - 2*\/ 6     3*\/  5 - 2*\/ 6   + \/ 6 *\/  5 - 2*\/ 6               3*\/  5 - 2*\/ 6   + \/ 6 *\/  5 - 2*\/ 6
$$\frac{\sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{2 \sqrt{6} + 5}} \right)}\right)}{\sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 3 \sqrt{5 - 2 \sqrt{6}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{5 - 2 \sqrt{6}}} \right)}}{\sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 3 \sqrt{5 - 2 \sqrt{6}}} + \frac{\pi}{\sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 3 \sqrt{5 - 2 \sqrt{6}}} + \frac{\pi \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5}}{\sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 3 \sqrt{5 - 2 \sqrt{6}}}$$ 
=
= 
                                                                /          ___   \                                                           _____________    _____________ /          /          ___   \\
                                                                |   -1 + \/ 2    |                                                          /         ___    /         ___  |          |   -1 + \/ 2    ||
                                                      -pi + atan|----------------|                                                        \/  5 - 2*\/ 6  *\/  5 + 2*\/ 6  *|-pi + atan|----------------||
                                                                |   _____________|                    _____________    _____________                                        |          |   _____________||
                                                                |  /         ___ |                   /         ___    /         ___                                         |          |  /         ___ ||
                     pi                                         \\/  5 - 2*\/ 6  /              pi*\/  5 - 2*\/ 6  *\/  5 + 2*\/ 6                                          \          \\/  5 + 2*\/ 6  //
------------------------------------------- + ------------------------------------------- + ------------------------------------------- + ----------------------------------------------------------------
     _____________            _____________        _____________            _____________        _____________            _____________                  _____________            _____________           
    /         ___      ___   /         ___        /         ___      ___   /         ___        /         ___      ___   /         ___                  /         ___      ___   /         ___            
3*\/  5 - 2*\/ 6   + \/ 6 *\/  5 - 2*\/ 6     3*\/  5 - 2*\/ 6   + \/ 6 *\/  5 - 2*\/ 6     3*\/  5 - 2*\/ 6   + \/ 6 *\/  5 - 2*\/ 6               3*\/  5 - 2*\/ 6   + \/ 6 *\/  5 - 2*\/ 6
$$\frac{\sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{2 \sqrt{6} + 5}} \right)}\right)}{\sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 3 \sqrt{5 - 2 \sqrt{6}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{5 - 2 \sqrt{6}}} \right)}}{\sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 3 \sqrt{5 - 2 \sqrt{6}}} + \frac{\pi}{\sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 3 \sqrt{5 - 2 \sqrt{6}}} + \frac{\pi \sqrt{5 - 2 \sqrt{6}} \sqrt{2 \sqrt{6} + 5}}{\sqrt{6} \sqrt{5 - 2 \sqrt{6}} + 3 \sqrt{5 - 2 \sqrt{6}}}$$ 
pi/(3*sqrt(5 - 2*sqrt(6)) + sqrt(6)*sqrt(5 - 2*sqrt(6))) + (-pi + atan((-1 + sqrt(2))/sqrt(5 - 2*sqrt(6))))/(3*sqrt(5 - 2*sqrt(6)) + sqrt(6)*sqrt(5 - 2*sqrt(6))) + pi*sqrt(5 - 2*sqrt(6))*sqrt(5 + 2*sqrt(6))/(3*sqrt(5 - 2*sqrt(6)) + sqrt(6)*sqrt(5 - 2*sqrt(6))) + sqrt(5 - 2*sqrt(6))*sqrt(5 + 2*sqrt(6))*(-pi + atan((-1 + sqrt(2))/sqrt(5 + 2*sqrt(6))))/(3*sqrt(5 - 2*sqrt(6)) + sqrt(6)*sqrt(5 - 2*sqrt(6)))
Respuesta numérica [src] 
0.604599788078073
0.604599788078073

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

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